Definite Integrals Questions with Solutions (Q40 – Q131) | Important for IIT JEE, CET, PET TGT PGT LT GRADE

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Complete set of Definite Integrals solved questions from Q40 to Q131 with step-by-step answers. Covers IIT JEE, CET, PET TGT PGT LT GRADE and competitive exam practice. Mobile-friendly, easy to read, and prepared in LaTeX format for clear maths rendering.

Definite Integrals — Q40–Q131

Definite Integrals — Questions Q40 → Q131

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Q40
Evaluate: \(\displaystyle \int_{\alpha}^{\beta} \dfrac{dx}{\sqrt{(x-\alpha)(\beta-x)}}\), \(\beta>\alpha\).
(a) 0
(b) \(\pi\)
(c) \(\beta-\alpha\)
(d) \(\dfrac{\pi}{\beta-\alpha}\)
Answer: (b)
Q41
Evaluate: \(\displaystyle \int_{\alpha}^{\beta} \sqrt{(x-\alpha)(\beta-x)}\,dx\).
(a) 0
(b) \(\dfrac{\pi^2}{2}(\beta^2-\alpha^2)\)
(c) \(\dfrac{\pi}{8}(\beta-\alpha)^2\)
(d) \(\dfrac{\pi^2}{16}(\beta-\alpha)\)
Answer: (c)
Q42
MP PET 2000
Evaluate: \(\displaystyle \int_{-1}^{1} \dfrac{dx}{x^{2}+2x+2}.\)
(a) 0
(b) \(\dfrac{\pi}{4}\)
(c) \(\dfrac{\pi}{2}\)
(d) \(-\dfrac{\pi}{4}\)
Answer: (b)
Q43
Punjab CEET 2001
Evaluate: \(\displaystyle \int_{1}^{e} \log x \, dx.\)
(a) 0
(b) 1
(c) e
(d) e-1
Answer: (b)
Q44
DCE 1999
Evaluate: \(\displaystyle \int_{1}^{x} \dfrac{\log (t^2)}{t}\,dt.\)
(a) \(\dfrac{\log x}{2}\)
(b) \((\log x)^2\)
(c) \(\tfrac{1}{2}(\log x)^2\)
(d) \((\log x)^2-1\)
Answer: (b)
Q45
Evaluate: \(\displaystyle \int_{0}^{\pi} \sin(mx)\sin(nx)\,dx\), where \(m\) and \(n\) are integers and \(m\neq n\).
(a) 0
(b) 1
(c) \(\dfrac{\pi}{2}\)
(d) \(\dfrac{1}{m+n}\)
Answer: (a)
Q46
JCECE 2002
Evaluate: \(\displaystyle \int_{2}^{4} x\sqrt{6-x}\,dx.\)
(a) \(\dfrac{25}{4}(2-\sqrt{3})\)
(b) \(\dfrac{27}{5}(2+\sqrt{5})\)
(c) \(\dfrac{32}{5}(3-\sqrt{2})\)
(d) \(\dfrac{41}{8}(5-\sqrt{2})\)
Answer: (c)
Q47
Evaluate: \(\displaystyle \int_{0}^{3} x\sqrt{1+x}\,dx.\)
(a) \(\dfrac{27}{15}\)
(b) \(\dfrac{112}{15}\)
(c) \(\dfrac{116}{15}\)
(d) \(\dfrac{128}{15}\)
Answer: (c)
Q48
Evaluate: \(\displaystyle \int_{0}^{1/\sqrt{3}} \dfrac{dx}{(1+x^2)\sqrt{1-x^2}}.\)
(a) \(\dfrac{\pi}{2}\)
(b) \(\dfrac{\pi}{\sqrt{2}}\)
(c) \(\dfrac{\pi}{2\sqrt{2}}\)
(d) \(\dfrac{\pi}{4\sqrt{2}}\)
Answer: (c)
Q49
KCET 2000
Evaluate: \(\displaystyle \int_{-1}^{3} \Big[ \tan^{-1}\!\big(\tfrac{x}{x^2+1}\big) + \tan^{-1}\!\big(\tfrac{x^2+1}{x}\big)\Big] dx.\)
(a) \(\pi\)
(b) \(2\pi\)
(c) \(3\pi\)
(d) \(4\pi\)
Answer: (b)
Q50
If \(f(x)\) is a monotonic differentiable function on \([a,b]\), then \(\displaystyle \int_{a}^{b} f(x)\,dx + \int_{f(a)}^{f(b)} f^{-1}(x)\,dx\) equals:
(a) \(bf(a) – af(b)\)
(b) \(bf(b) – af(a)\)
(c) f(a)+f(b)
(d) Cannot be determined
Answer: (b)
Q51
UPSEAT 2003
Evaluate: \(\displaystyle \int_{8}^{15} \dfrac{dx}{(x-3)\sqrt{x+1}}.\)
(a) \(\tfrac{1}{2}\log\!\left(\tfrac{5}{3}\right)\)
(b) \(\tfrac{1}{3}\log\!\left(\tfrac{2}{3}\right)\)
(c) \(\tfrac{1}{5}\log\!\left(\tfrac{3}{5}\right)\)
(d) \(\tfrac{2}{3}\log\!\left(\tfrac{2}{5}\right)\)
Answer: (a)
Q52
Evaluate: \(\displaystyle \int_{0}^{1}\dfrac{\log 5\,e^{x}\sqrt{e^{x}-1}}{e^{x}+3}\,dx.\) (scanned expression)
(a) \(2+\pi\)
(b) \(4-\pi\)
(c) \(\pi-5\)
(d) \(3+2\pi\)
Answer: (b)
Q53
AMU 1994
Let \(I_{m,n}=\displaystyle\int_{0}^{1} x^{m}(\ln x)^{n}\,dx.\) Then \(I_{m,n}\) is equal to:
(a) \(\dfrac{n}{n+1}I_{m,n-1}\)
(b) \(-\dfrac{m}{n+1}I_{m,n-1}\)
(c) \(-\dfrac{n}{m+1}I_{m,n-1}\)
(d) \(-\dfrac{m}{n+1}I_{m,n-1}\)
Answer: (c)
Q54
Punjab CEET 2002
Evaluate: \(\displaystyle \int_{0}^{1} x^{2} e^{x}\,dx.\)
(a) \(e-2\)
(b) \(e^{2}\)
(c) \(e^{2}-2\)
(d) \(e+4\)
Answer: (a)
Q55
DCE 2005
Let \(f(x)\) be a function satisfying \(f'(x)=f(x)\) with \(f(0)=1\) and \(g(x)\) be a function that satisfies \(f(x)+g(x)=x^2\). Then value of \(\displaystyle\int_{0}^{1} f(x)g(x)\,dx\) equals:
(a) \(e + \dfrac{e^2}{2} + \dfrac{5}{2}\)
(b) \(e – \dfrac{e^2}{2} – \dfrac{5}{2}\)
(c) \(e + \dfrac{e^2}{2} – \dfrac{3}{2}\)
(d) \(e – \dfrac{e^2}{2} – \dfrac{3}{2}\)
Answer: (d)
Q56
Evaluate: \(\displaystyle \int_{1/2}^{2} \dfrac{1}{x}\sin\!\Big(x-\dfrac{1}{x}\Big)\,dx.\)
(a) 0
(b) 1
(c) \(\dfrac{\pi}{2}\)
(d) \(\pi\)
Answer: (a)
Q57
EAMCET 1995
Evaluate: \(\displaystyle \int_{0}^{\infty} \big(a^{-x}-b^{-x}\big)\,dx\) where \(a>1, b>1.\)
(a) \(\dfrac{\log(b/a)}{\log(ab)}\)
(b) \(\dfrac{1}{\log a}-\dfrac{1}{\log b}\)
(c) \(\log\!\left(\dfrac{b}{a}\right)\)
(d) \(\dfrac{a\log b}{b\log a}\)
Answer: (b)
Q58
KCET 1997
If \(\displaystyle \int \dfrac{dx}{a+b\cos x}=\dfrac{1}{\sqrt{a^2-b^2}}\cos^{-1}\!\Big(\dfrac{b+a\cos x}{a+b\cos x}\Big)\), then evaluate \(\displaystyle \int_{0}^{\pi} \dfrac{dx}{a+b\cos x}\).
(a) \(\dfrac{\pi}{2\sqrt{b^2-a^2}}\)
(b) \(\dfrac{1}{\sqrt{a^2-b^2}}\)
(c) \(\dfrac{\pi}{\sqrt{a^2-b^2}}\)
(d) \(\dfrac{2ab}{\sqrt{a^2-b^2}}\)
Answer: (c)
Q59
AMU 1995
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \sqrt{\cos x – \cos^{3}x}\,dx.\)
(a) \(\tfrac{1}{2}\)
(b) \(\tfrac{2}{3}\)
(c) \(\tfrac{3\pi}{4}\)
(d) \(\pi – \tfrac{1}{2}\)
Answer: (b)
Q60
DCE 2001
Evaluate: \(\displaystyle \int_{1}^{e} \dfrac{e^{-1/x}}{x^{2}}\,dx.\)
(a) \(\dfrac{1}{\sqrt{e}}+\dfrac{1}{e}\)
(b) \(\dfrac{1}{2\sqrt{e}}-\dfrac{2}{e}\)
(c) \(\dfrac{1}{\sqrt{e}}-\dfrac{1}{e}\)
(d) \(\dfrac{2}{\sqrt{e}}-\dfrac{1}{2e}\)
Answer: (c)
Q61
EAMCET 2003
Evaluate: \(\displaystyle \int_{0}^{3} \dfrac{3x+1}{x^{2}+9}\,dx.\)
(a) \(\log(2\sqrt{3})+\dfrac{\pi}{8}\)
(b) \(\log(2\sqrt{2})+\dfrac{\pi}{12}\)
(c) \(\log(3\sqrt{2})+\dfrac{\pi}{16}\)
(d) \(\log(3\sqrt{3})+\dfrac{\pi}{6}\)
Answer: (b)
Q62
KCET 2002 / Haryana CEET 1997
Evaluate: \(\displaystyle \int_{0}^{\pi} \dfrac{dx}{5+3\cos x}.\)
(a) 0
(b) \(\dfrac{\pi}{2}\)
(c) \(\dfrac{\pi}{4}\)
(d) \(\dfrac{\pi}{8}\)
Answer: (c)
Q63
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \dfrac{dx}{4+5\cos x}.\)
(a) \(-\tfrac{1}{3}\log 2\)
(b) \(\tfrac{1}{3}\log 2\)
(c) \(\tfrac{1}{9}\log 2\)
(d) \(\tfrac{1}{6}\log 2\)
Answer: (b)
Q64
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \dfrac{dx}{2+\cos x}.\)
(a) \(\dfrac{\pi}{6\sqrt{3}}\)
(b) \(\dfrac{2\pi}{\sqrt{3}}\)
(c) \(\dfrac{\pi}{3\sqrt{3}}\)
(d) \(\dfrac{\pi}{\sqrt{3}}\)
Answer: (c)
Q65
Evaluate: \(\displaystyle \int_{0}^{1} (tx+1-x)^{n} dx.\) (n integer)
(a) \(\dfrac{t^{n}+1}{n+1}\)
(b) \(\dfrac{t^{n+1}-1}{n+1}\)
(c) \(\dfrac{t^{n+1}-1}{(n-1)(t-1)}\)
(d) \(\dfrac{(t^{n}+t^{n-1}+\dots+1)}{(n+1)}\)
Answer: (d)
Q66
EAMCET 2002
Evaluate: \(\displaystyle \int_{2}^{3} \dfrac{dx}{x^{2}-x}.\)
(a) \(\log \tfrac{1}{2}\)
(b) \(\log \tfrac{1}{3}\)
(c) \(\log \dfrac{4}{3}\)
(d) \(\log \dfrac{3}{8}\)
Answer: (c)
Q67
UPSEAT 2003
If \(\displaystyle \int_{0}^{\pi/3} \dfrac{\cos x}{3+4\sin x}\,dx = k\log\!\left(\dfrac{3+2\sqrt{3}}{3}\right)\), then find \(k\).
(a) \(\dfrac{3}{2}\)
(b) \(\dfrac{7}{3}\)
(c) \(\dfrac{1}{4}\)
(d) \(\dfrac{2}{5}\)
Answer: (c)
Q68
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \dfrac{dx}{2\cos x + \sin x}.\)
(a) \(\dfrac{2}{\sqrt{5}}\log\tfrac{3}{2}\)
(b) \(\dfrac{2}{\sqrt{5}}\log\!\left(\dfrac{\sqrt{3}+2}{\sqrt{3}-2}\right)\)
(c) \(\dfrac{2}{\sqrt{5}}\log\!\left(\dfrac{\sqrt{5}+1}{\sqrt{5}-1}\right)\)
(d) \(\dfrac{2}{\sqrt{5}}\log\!\left(\dfrac{2-\sqrt{5}}{2+\sqrt{5}}\right)\)
Answer: (c)
Q69
DCE 1997
Evaluate: \(\displaystyle \int_{0}^{1} \log\big(\sqrt{1-x}+\sqrt{1+x}\big)\,dx.\)
(a) \(\tfrac{1}{2}\big[\log 2 – \tfrac{\pi}{4} + 1\big]\)
(b) \(\tfrac{1}{2}\big[\log 2 + \tfrac{\pi}{2} -1\big]\)
(c) \(\tfrac{1}{2}\big[\log 2 – \tfrac{3\pi}{2} -1\big]\)
(d) \(\tfrac{1}{2}\big[\log 2 + \tfrac{\pi}{3} +1\big]\)
Answer: (b)
Q70
EAMCET 1999
Evaluate: \(\displaystyle \int_{0}^{\pi/4} (\tan^{4}x + \tan^{2}x)\,dx.\)
(a) \(\tfrac{1}{3}\)
(b) \(\tfrac{2}{3}\)
(c) \(\tfrac{4}{3}\)
(d) \(\tfrac{5}{3}\)
Answer: (a)
Q71
KCET 2001
Evaluate: \(\displaystyle \int_{0}^{1} \dfrac{\tan^{-1} x}{1+x^{2}}\,dx.\)
(a) \(\dfrac{\pi}{4}\)
(b) \(\dfrac{\pi^{2}}{32}\)
(c) \(\dfrac{\pi^{2}-2}{16}\)
(d) \(\dfrac{2\pi^{2}-1}{8}\)
Answer: (b)
Q72
The solution of the equation \(\displaystyle \int_{\sqrt{2}}^{x} \dfrac{1}{t\sqrt{t^{2}-1}}\,dt = \dfrac{\pi}{2}\) is:
(a) \(x=1\)
(b) \(x=+\sqrt{2}\)
(c) \(x=2\sqrt{2}\)
(d) \(x=3\sqrt{2}\)
Answer: (c)
Q73
Gujarat CET 2006
If \(\displaystyle \int_{\sqrt{2}}^{\alpha} \dfrac{dx}{x\sqrt{x^{2}-1}} = \dfrac{\pi}{12}\), then \(\alpha\) = ?
(a) 2
(b) 4
(c) \(2\sqrt{2}\)
(d) \(\tfrac{2}{\sqrt{3}}\)
Answer: (a)
Q74
EAMCET 1998 / J&K CET 2007
Evaluate: \(\displaystyle \int_{0}^{a} \sqrt{a^{2}-x^{2}}\,dx.\)
(a) \(\pi a^{2}/2\)
(b) \(\pi a^{2}/4\)
(c) \(\pi a^{2}/8\)
(d) \(\pi a\sqrt{a}/8\)
Answer: (b)
Q75
Roorkee 1990
Evaluate: \(\displaystyle \int_{0}^{\infty} \dfrac{dx}{(x+\sqrt{1+x^{2}})^{n}},\; n>1.\)
(a) \(\tfrac{1}{n+1}\)
(b) \(\tfrac{n+1}{n-1}\)
(c) \(\tfrac{n}{n^{2}-1}\)
(d) \(\tfrac{n^{2}-1}{n^{2}+1}\)
Answer: (c)
Q76
EAMCET 1992
Evaluate: \(\displaystyle \int_{0}^{\infty} \dfrac{dx}{(x+\sqrt{1+x^{2}})^{3}}.\)
(a) 1/4
(b) 1/8
(c) 1/2
(d) 3/8
Answer: (b)
Q77
MP PET 2003
Evaluate: \(\displaystyle \int_{0}^{1} \dfrac{\cos x}{(1+\sin x)(2+\sin x)}\,dx.\)
(a) \(\log \tfrac{2}{3}\)
(b) \(\log \tfrac{2}{5}\)
(c) \(\log \tfrac{4}{3}\)
(d) \(\log \tfrac{3}{5}\)
Answer: (b)
Q78
Evaluate: \(\displaystyle \int_{0}^{1} \cos \bigg(2\cot^{-1}\sqrt{\tfrac{1-x}{1+x}}\bigg)\,dx.\)
(a) –1/2
(b) –1
(c) 3/4
(d) 3/8
Answer: (a)
Q79
EAMCET 1996
Evaluate: \(\displaystyle \int_{-\pi/2}^{\pi/2} \cos 3x(1+\sin x)^{2}\,dx.\)
(a) –1/15
(b) –2/3
(c) –8/5
(d) –15/14
Answer: (c)
Q80
Evaluate: \(\displaystyle \int_{2}^{4} \dfrac{\sqrt{x^{2}-4}}{x}\,dx.\)
(a) \(\tfrac{\sqrt{3}-\pi}{2\sqrt{2}}\)
(b) \(\tfrac{2}{3}(3\sqrt{3}-\pi)\)
(c) \(\tfrac{1}{3}(2\sqrt{2}-\pi)\)
(d) \(2\sqrt{3}-\sqrt{2}\pi\)
Answer: (b)
Q81
EAMCET 1995
Evaluate: \(\displaystyle \int_{0}^{1} x\tan^{-1}x\,dx.\)
(a) \(\tfrac{\pi}{2}-\tfrac{1}{4}\)
(b) \(\tfrac{\pi}{6}-\tfrac{1}{2}\)
(c) \(\tfrac{\pi}{4}-\tfrac{1}{2}\)
(d) \(\tfrac{\pi}{8}-\tfrac{1}{4}\)
Answer: (c)
Q82
MP PET 1994
Evaluate: \(\displaystyle \int_{-\alpha}^{\alpha} f(x)\,dx.\)
(a) 0
(b) \(\int_{0}^{\alpha} f(x)\,dx\)
(c) \(2\int_{0}^{\alpha} f(x)\,dx\)
(d) \(\int_{0}^{\alpha} [f(x)+f(-x)]\,dx\)
Answer: (d)
Q83
KCET 1999
Evaluate: \(\displaystyle \int_{0}^{1} \sin^{-1}\!\Big(\tfrac{2x}{1+x^{2}}\Big)\,dx.\)
(a) \(\tfrac{\pi}{2}-2\log 2\)
(b) \(\tfrac{\pi}{4}+\log 2\)
(c) \(\tfrac{\pi}{2}-\log 2\)
(d) \(\tfrac{\pi}{2}-2\log 2\)
Answer: (c)
Q84
EAMCET 1996
Evaluate: \(\displaystyle \int_{0}^{\pi/2} (\cos x-\sin x)e^{x}\,dx.\)
(a) 0
(b) 1
(c) –1
(d) –e
Answer: (c)
Q85
DCE 2001
The function \(L(x)=\int_{1}^{x}\tfrac{dt}{t}\) satisfies which relation?
(a) \(L(x+y)=L(x)+L(y)\)
(b) \(L(\tfrac{x}{y})=L(x)+L(y)\)
(c) \(L(xy)=L(x)+L(y)\)
(d) \(L(x-y)=L(x)-L(y)\)
Answer: (c)
Q86
EAMCET 1996
Evaluate: \(\displaystyle \int_{0}^{1} \dfrac{1-x}{1+x}\,dx.\)
(a) \(\log 2-1\)
(b) \(2\log 2-1\)
(c) \(\log(3/2)-1\)
(d) \(\log(2/3)-1\)
Answer: (b)
Q87
Haryana CEET 2002
Evaluate: \(\displaystyle \int_{0}^{1} \dfrac{dx}{x^{2}+2x\cos\alpha+1}.\)
(a) \(\tfrac{1}{2\alpha\sin\alpha}\)
(b) \(\alpha\sin\alpha\)
(c) \(\tfrac{\alpha}{2\sin\alpha}\)
(d) \(\tfrac{\alpha}{2}\sin\alpha\)
Answer: (c)
Q88
Evaluate: \(\displaystyle \int_{0}^{\pi} \cos^{2}\!\alpha \cos^{2}x\,dx.\)
(a) \(2\pi/(a^{2}-4)\)
(b) \(\pi a \log 2 /4\)
(c) \(\pi/(1-a^{2})\)
(d) \(4\pi/(2-a^{2})\)
Answer: (c)
Q89
DCE 1997
If \(I_{n}=\int_{0}^{\pi/2} x^{n}\sin x\,dx,\) then value of \(I_{4}+12I_{2}=?\)
(a) 4π
(b) \((3π/2)^3\)
(c) \((π/2)^2\)
(d) \(4(π/2)^3\)
Answer: (d)
Q90
If h(a)=h(b), then value of \(\displaystyle \int_{a}^{b}[f(g(h(x)))]^{-1}f'(g(h(x)))g'(h(x))h'(x)\,dx.\)
(a) 0
(b) b–a
(c) f(b)–f(a)
(d) f(g(b))-f(g(a))
Answer: (c)
Q91
EAMCET 1996
Evaluate: \(\displaystyle \int_{0}^{1}\dfrac{dx}{x+\sqrt{x}}.\)
(a) 1
(b) 1/√2
(c) 2\log 2
(d) \(\log(2+\sqrt{2})\)
Answer: (d)
Q92
Punjab CEET 2001
Evaluate: \(\displaystyle \int_{0}^{3} \dfrac{2-3x}{\sqrt{1+x}}\,dx.\)
(a) 2\log(2/3^{2/3})
(b) \log(2/√3)
(c) 4\log(3/2^3)
(d) 2\log(2/3e^3)
Answer: (a)
Q93
Evaluate: \(\displaystyle \int_{0}^{16} \tan^{-1}\!\big(\sqrt{\sqrt{x}-1}\big)\,dx.\)
(a) \(\tfrac{4\pi}{15}-3\sqrt{3}\)
(b) \(\tfrac{16\pi}{3}-2\sqrt{3}\)
(c) \(\tfrac{9\pi}{16}-2\sqrt{2}\)
(d) \(\tfrac{3\pi}{10}-3\sqrt{2}\)
Answer: (b)
Q94
EAMCET 1996
Evaluate: \(\displaystyle \int_{0}^{\pi/4} \dfrac{\sin^{9}x}{\cos^{11}x}\,dx.\)
(a) 1/2
(b) 1/5
(c) 1/10
(d) 1/15
Answer: (c)
Q95
Orissa JEE 2005, MP PET 2001
Evaluate: \(\displaystyle \int_{0}^{1} \sin^{2}(\sin^{-1}\sqrt{t})\,dt + \int_{0}^{1} \cos^{2}(\cos^{-1}\sqrt{t})\,dt.\)
(a) 1
(b) \(\pi/2\)
(c) \(\pi/4\)
(d) \(\sqrt{\pi}/2\)
Answer: (c)
Q96
Jamia Millia Islamia 2003
If \(f(3-x)=f(x)\), then \(\displaystyle \int_{1}^{2} x f(x)\,dx\) is equal to:
(a) \(\tfrac{1}{2}\int_{1}^{2} f(x)\,dx\)
(b) \(\tfrac{3}{2}\int_{1}^{2} f(2-x)\,dx\)
(c) \(\tfrac{1}{2}\int_{1}^{2} f(3-x)\,dx\)
(d) \(\tfrac{3}{2}\int_{1}^{2} f(x)\,dx\)
Answer: (d)
Q97
Roorkee 1995
Evaluate: \(\displaystyle \int_{0}^{1} \dfrac{2-x^{2}}{(1+x)\sqrt{1-x^{2}}}\,dx.\)
(a) \(\pi/6\)
(b) \(\pi/3\)
(c) \(\pi/2\)
(d) \(\pi/4\)
Answer: (c)
Q98
Punjab CEET 2002
If \(g(x)=\int_{0}^{x} \cos^{4}t\,dt\), then \(g(x+\pi)\) equals:
(a) g(x)+g(π)
(b) g(x)–g(π)
(c) g(x)g(π)
(d) g(x)/g(π)
Answer: (a)
Q99
EAMCET 1998
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \big(2\tan \tfrac{x}{2}+x\sec^{2}\tfrac{x}{2}\big)\,dx.\)
(a) 0
(b) \(\pi/4\)
(c) \(\pi/2\)
(d) \(\pi\)
Answer: (d)
Q100
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \dfrac{x+\sin x}{1+\cos x}\,dx.\)
(a) π
(b) 2π
(c) \(\pi/2\)
(d) \(\pi^{2}/2\)
Answer: (c)
Q101
For each nonzero x, let \(f(1/x)+x^{2}f(x)=0.\) Then \(\displaystyle \int \dfrac{\csc \theta}{\sin\theta}f(x)\,dx = ?\)
(a) 0
(b) \(\csc\theta-\sin\theta\)
(c) \(\sin^{2}\theta\)
(d) \(\csc^{2}\theta\)
Answer: (a)
Q102
Kurukshetra 1996
Evaluate: \(\displaystyle \int_{0}^{\pi/4} \dfrac{\sqrt{\tan x}}{\sin x \cos x}\,dx.\)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (c)
Q103
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \dfrac{\sin^{2}x}{(1+\cos x)^{2}}\,dx.\)
(a) \(\pi/2\)
(b) \(\pi/4\)
(c) \(\pi/2 – 1\)
(d) \(\pi/4 – \pi^{2}\)
Answer: (c)
Q104
AMU 2003
The values of \(\alpha\) which satisfy \(\int_{0}^{\pi/2} \sin x\,dx = \sin 2\alpha\), where \(\alpha \in (0,2\pi)\) are:
(a) \(\pi/2\)
(b) \(3\pi/2\)
(c) \(7\pi/6\)
(d) All of these
Answer: (d)
Q105
Evaluate: \(\displaystyle \int_{0}^{\pi/2} \dfrac{\sin 2x}{a^{2}+b^{2}\sin^{2}x}\,dx.\)
(a) 0
(b) \(\pi/4\)
(c) \(\tfrac{1}{b^{2}}\log\Big(\dfrac{a^{2}+b^{2}}{a^{2}}\Big)\)
(d) \(\dfrac{a^{2}}{b^{2}}\log\big(a^{2}+b^{2}\big)\)
Answer: (d)
Q106
Evaluate: \(\displaystyle \int_{0}^{\pi^{2}/4} \sin\sqrt{x}\,dx.\)
(a) 2
(b) –2
(c) \(\pi+2\)
(d) \(\pi-2\)
Answer: (a)
Q107
Haryana CEET 1995
Evaluate: \(\displaystyle \int_{0}^{1} x(1-x)^{4}\,dx.\)
(a) 0
(b) 1/45
(c) 1/30
(d) 1/15
Answer: (d)
Q108
If \(\displaystyle \int_{1}^{\alpha} (\alpha-4x)\,dx \geq 6-5\alpha,\; \alpha>1\), then \(\alpha=\) ?
(a) 3/2
(b) 6/5
(c) 2
(d) 12/5
Answer: (c)
Q109
Roorkee 1982
Evaluate: \(\displaystyle \int_{0}^{1} \dfrac{dx}{(x^{2}+1)^{3/2}}.\)
(a) 1/(2√2)
(b) 1/√2
(c) √2
(d) 2√2
Answer: (c)
Q110
Evaluate: \(\displaystyle \int_{0}^{\pi/4} \dfrac{x\sin x}{\cos^{3}x}\,dx.\)
(a) (π–1)/4
(b) (π/4–1/2)
(c) (π/2–1)
(d) (3π–2)/4
Answer: (c)
Q111
Roorkee 1982
Evaluate: \(\displaystyle \int_{0}^{2\pi} e^{x/2}\sin\!\big(\tfrac{x}{2}+\pi/4\big)\,dx.\)
(a) 2√2
(b) 2π
(c) e^{\pi}
(d) 0
Answer: (c)
Q112
MP PET 1997
Evaluate: \(\displaystyle \int_{0}^{1} \tan^{-1}x\,dx.\)
(a) \(\pi/4+2\log 2\)
(b) \(\pi/2+\log 2\)
(c) \(\pi/4-\log 2\)
(d) \(\pi/4-(1/2)\log 2\)
Answer: (b)
Q113
DCE 1995
Evaluate: \(\displaystyle \int_{0}^{3\alpha} \csc(x-\alpha)\csc(x-2\alpha)\,dx.\)
(a) \(2\csc \alpha \log\!\big(\tfrac{1}{2}\sec \alpha\big)\)
(b) \(2\csc 2\alpha \log\!\big(\tfrac{1}{2}\csc \alpha\big)\)
(c) \(2\csc \alpha \log\!\big(\tfrac{1}{2}\tan \alpha\big)\)
(d) \(2\csc 2\alpha \log\!\big(\tfrac{1}{2}\cot \alpha\big)\)
Answer: (a)
Q114
Evaluate: \(\displaystyle \int_{-π/4}^{π/4} e^{-x}\sin x\,dx.\)
(a) (1/√2)(e^{-π/4}-1)
(b) –(1/√2)e^{-π/4}
(c) –√2(e^{-π/4}-e^{π/4})
(d) 0
Answer: (b)
Q115
IIT 1997
Evaluate: \(\displaystyle \int_{1}^{e} \dfrac{\sin(x\log e^x)}{x}\,dx.\)
(a) 0
(b) 2
(c) 37
(d) 74
Answer: (b)
Q116
Evaluate: \(\displaystyle \int_{0}^{2} \sqrt{\dfrac{2+x}{2-x}}\,dx.\)
(a) π/2+1
(b) π+2
(c) √π
(d) √(2π)
Answer: (b)
Q117
Evaluate: \(\displaystyle \int_{0}^{1} \dfrac{x^{3}}{(1+x^{8})}\,dx.\)
(a) π/2
(b) π/4
(c) π/8
(d) π/16
Answer: (d)
Q118
Evaluate: \(\displaystyle \int_{2}^{4} \dfrac{\sqrt{x^{2}-4}}{x^{4}}\,dx.\)
(a) 2√3/27
(b) 3√3/16
(c) √3/32
(d) √6/28
Answer: (c)
Q119
Evaluate: \(\displaystyle \int_{1/3}^{1} \dfrac{(x-x^{3})^{1/3}}{x^{4}}\,dx.\)
(a) 0
(b) 3
(c) 6
(d) 9
Answer: (c)
Q120
Evaluate: \(\displaystyle \int_{0}^{1} \sqrt{x(1-x)}\,dx.\)
(a) 2√2
(b) √2
(c) π/8
(d) π/16
Answer: (b)
Q121
Evaluate: \(\displaystyle \int_{0}^{1/2} \dfrac{dx}{\sqrt{x-x^{2}}}.\)
(a) 2√π
(b) π/2
(c) π–3/2
(d) π²/4
Answer: (b)
Q122
Evaluate: \(\displaystyle \int_{0}^{4} \dfrac{dx}{(1+\sqrt{x})}.\)
(a) 4–2\log 3
(b) 2–3\log 3
(c) 2–3\log 2
(d) 4–\log 6
Answer: (a)
Q123
If \(\int_{0}^{\log2} \dfrac{dx}{e^x+1} = \log(3/2),\) then \(\alpha = ?\)
(a) e^{2}-2
(b) \(\log 2\)
(c) \(\log(3/2)\)
(d) 2\log 2
Answer: (a)
Q124
Orissa JEE 2005
If \(\int_{0}^{\log 2} \dfrac{dx}{\sqrt{e^x-1}} = \pi/6\), then \(\alpha=\ ?\)
(a) 1/√e
(b) √2 – 2
(c) \log 2
(d) 2 \log 2
Answer: (c)
Q125
Haryana CEET 2000
Evaluate: \(\displaystyle \int_{\pi/6}^{\pi/3} \dfrac{dx}{\sin 2x}.\)
(a) \log 3
(b) \log \sqrt{3}
(c) (1/3) \log 2
(d) \log \sqrt{6}
Answer: (d)
Q126
Evaluate: \(\displaystyle \int_{0}^{\infty} \dfrac{dx}{1+x^4}.\)
(a) \(\pi/\sqrt{2}\)
(b) (π–2)/√2
(c) \(\pi/(2\sqrt{2})\)
(d) (2π–1)/√2
Answer: (c)
Q127
Let \(I_1 = \int_{0}^{1} \dfrac{dt}{1+t^2},\ I_2=\int_{1}^{1/x}\dfrac{dt}{1+t^2}, x>0.\) Then,
(a) \(I_1=I_2\)
(b) \(I_1
(c) \(I_1>I_2\)
(d) \(I_1+I_2=0\)
Answer: (a)
Q128
If \(\alpha \in (-π,0)\) such that \(\sin α+ \int_{0}^{2α}\cos 2x\,dx=0\), then
(a) α = –π/2
(b) α = –π/3
(c) α = –π/6
(d) α = –π/12
Answer: (b)
Q129
Evaluate: \(\displaystyle \int_{1/e}^{\tan x} \dfrac{t}{1+t^2}\,dt + \int_{1/e}^{\cot x} \dfrac{dt}{1+t^2}.\)
(a) 0
(b) 1
(c) e
(d) 1/e
Answer: (c)
Q130
The values of \(\alpha\) for which \(\int_{0}^{\alpha} (3x^2+4x–5)\,dx \leq \alpha^3–2\) are given by:
(a) α ≤ 1/2
(b) α ≥ 2
(c) 1/2 ≤ α ≤ 2
(d) values of α not real
Answer: (c)
Q131
How many positive integral values of α satisfy the inequality: \(\int_{0}^{\pi/2}\Big[\tfrac{α^2}{4}(\cos 3x – \tfrac{3}{4}\cos x)+α\sin x-20\cos x\Big]dx \leq -α^2/3 ?\)
(a) One
(b) Two
(c) Four
(d) Ten
Answer: (b)
यह पूरा HTML Q40–Q131 का सेट है — MathJax सक्रिय है। अगर तुम चाहो तो मैं अब यह फाइल ZIP में रखकर दे दूँ या किसी specific format (PDF / DOCX) में export करके दे दूँ। और कोई बदलाव चाहिए (फॉन्ट, रंग, answer visibility hide/show, print-मैथ के लिए tweaks) बताओ — मैं तुरंत apply कर दूँगा।

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