SHORT CUT CALCULUS FORMULA

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SHORT CUT CALCULUS FORMULA

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INDEFINITE INTEGRAL — Full Notes

Integration: It is the inverse process of differentiation. If the derivative of \(F(x)\) is \(f(x)\), then \(F\) is an antiderivative of \(f\) and we write

\[ \frac{d}{dx}\{F(x)\}=f(x)\quad\Rightarrow\quad \int f(x)\,dx = F(x)+C . \]

Moreover, if \(\frac{d}{dx}\{F(x)\}=f(x)\), then for any constant \(C\), \(\frac{d}{dx}\{F(x)+C\}=f(x)\). Thus a given function may have infinitely many antiderivatives that differ by a constant \(C\); these are called indefinite integrals.

Properties of Indefinite Integrals

\[ \text{(i)}\;\; \int\!\{f(x)+g(x)\}\,dx=\int f(x)\,dx+\int g(x)\,dx \] \[ \text{(ii)}\;\; \int \frac{d}{dx}[F(x)]\,dx = F(x)+C \] \[ \text{(iii)}\;\; \int k\,f(x)\,dx = k\int f(x)\,dx \] \[ \text{(iv)}\;\; \int\!\big[k_1 f_1(x)\pm k_2 f_2(x)\pm\cdots\pm k_n f_n(x)\big]dx =k_1\!\int f_1(x)dx \pm \cdots \pm k_n\!\int f_n(x)dx + C \] \[ \text{(v)}\;\; \int \frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C \] \[ \text{(vi)}\;\; \text{If }\int f(x)\,dx=F(x),\text{ then }\int f(ax+b)\,dx=\frac{1}{a}F(ax+b)+C,\; a\ne0 . \]

Formulae of Indefinite Integrals (based on definition)

1. \(\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C,\; (n\neq-1)\)

2. \(\displaystyle \int \frac{1}{x}\,dx=\ln|x|+C\)

3. \(\displaystyle \int e^{x}dx=e^{x}+C\)

4. \(\displaystyle \int a^{x}dx=\frac{a^{x}}{\ln a}+C,\; a>0,a\neq1\)

5. \(\displaystyle \int \cos x\,dx=\sin x+C\)

6. \(\displaystyle \int \sin x\,dx=-\cos x+C\)

7. \(\displaystyle \int \sec^{2}x\,dx=\tan x+C\)

8. \(\displaystyle \int \csc^{2}x\,dx=-\cot x+C\)

9. \(\displaystyle \int \sec x\tan x\,dx=\sec x+C\)

10. \(\displaystyle \int \csc x\cot x\,dx=-\csc x+C\)

11. \(\displaystyle \int \frac{dx}{\sqrt{1-x^{2}}}= \sin^{-1}x + C\)

12. \(\displaystyle \int \frac{dx}{1+x^{2}}= \tan^{-1}x + C\)

13. \(\displaystyle \int \frac{dx}{\sqrt{x^{2}-1}} = \ln\!\big|x+\sqrt{x^{2}-1}\big| + C = \cosh^{-1}x + C\)

14. \(\displaystyle \int \frac{dx}{x\sqrt{x^{2}-1}}= \sec^{-1}|x| + C\)

15. \(\displaystyle \int \cosh x\,dx= \sinh x + C\)

16. \(\displaystyle \int \sinh x\,dx= \cosh x + C\)

17. \(\displaystyle \int \sech^{2}x\,dx= \tanh x + C\)

18. \(\displaystyle \int \csch^{2}x\,dx= -\coth x + C\)

19. \(\displaystyle \int \sech x \tanh x\,dx= -\sech x + C,\quad \int \csch x \coth x\,dx= -\csch x + C\)

Other Derived Formulae

20. \(\displaystyle \int \tan x\,dx = -\ln|\cos x| + C = \ln|\sec x| + C\)

21. \(\displaystyle \int \cot x\,dx= \ln|\sin x| + C\)

22. \(\displaystyle \int \sec x\,dx = \ln|\sec x+\tan x| + C = \ln\!\left|\tan\!\left(\frac{\pi}{4}+\frac{x}{2}\right)\right| + C\)

23. \(\displaystyle \int \csc x\,dx = \ln|\csc x-\cot x| + C = \ln\!\left|\tan\!\left(\frac{x}{2}\right)\right| + C\)

24. \(\displaystyle \int \frac{dx}{x^{2}-a^{2}} = \frac{1}{2a}\,\ln\left|\frac{x-a}{x+a}\right| + C\)

25. \(\displaystyle \int \frac{dx}{a^{2}-x^{2}} = \frac{1}{2a}\,\ln\left|\frac{a+x}{a-x}\right| + C\)

26. \(\displaystyle \int \frac{dx}{x^{2}+a^{2}} = \frac{1}{a}\tan^{-1}\!\left(\frac{x}{a}\right) + C\)

27. \(\displaystyle \int \frac{dx}{\sqrt{x^{2}+a^{2}}} = \sinh^{-1}\!\left(\frac{x}{a}\right) + C = \ln\!\big|x+\sqrt{x^{2}+a^{2}}\big| + C\)

28. \(\displaystyle \int \frac{dx}{\sqrt{x^{2}-a^{2}}} = \cosh^{-1}\!\left(\frac{x}{a}\right) + C = \ln\!\big|x+\sqrt{x^{2}-a^{2}}\big| + C\)

29. \(\displaystyle \int \frac{dx}{\sqrt{a^{2}-x^{2}}} = \sin^{-1}\!\left(\frac{x}{a}\right) + C\)

30. \(\displaystyle \int \sqrt{x^{2}-a^{2}}\,dx = \frac{x}{2}\sqrt{x^{2}-a^{2}} – \frac{a^{2}}{2}\ln\!\big|x+\sqrt{x^{2}-a^{2}}\big| + C\)

31. \(\displaystyle \int \sqrt{a^{2}-x^{2}}\,dx = \frac{x}{2}\sqrt{a^{2}-x^{2}} + \frac{a^{2}}{2}\sin^{-1}\!\left(\frac{x}{a}\right) + C\)

32. \(\displaystyle \int \sqrt{x^{2}+a^{2}}\,dx = \frac{x}{2}\sqrt{x^{2}+a^{2}} + \frac{a^{2}}{2}\ln\!\big|x+\sqrt{x^{2}+a^{2}}\big| + C\)

Tip: \(\int f(ax+b)\,dx=\frac1a F(ax+b)+C\) and always verify by differentiating your result.

Integrals: Full Rules & Worked Types (Readable, Mobile)

Integrals — Full Page Formula Sheet (Mobile + WordPress Friendly)

Saare rules MathJax me type kiye gaye hain taaki equations saaf dikhein. Desktop par 2-column, mobile par 1-column responsive layout.

A. Special Results (33–38)

33. \(\displaystyle \int e^{x}\,[\,f(x)+f'(x)\,]\,dx \;=\; e^{x}f(x)+C.\)

Reason: \(\dfrac{d}{dx}\big(e^{x}f(x)\big)=e^{x}f(x)+e^{x}f'(x).\)

34. \(\displaystyle \int e^{kx}\,[\,k\,f(x)+f'(x)\,]\,dx \;=\; e^{kx}f(x)+C\qquad (k\in\mathbb{R}).\)

Because \(\dfrac{d}{dx}\big(e^{kx}f(x)\big)=ke^{kx}f(x)+e^{kx}f'(x).\)

35. \(\displaystyle \int e^{g(x)}\,[\,g'(x)\,f(x)+f'(x)\,]\,dx \;=\; e^{g(x)}\,f(x)+C.\)

Generalization: \(\dfrac{d}{dx}\big(e^{g(x)}f(x)\big)=e^{g(x)}\big(g'(x)f(x)+f'(x)\big).\)

36. \(\displaystyle \int e^{ax}\sin(bx+c)\,dx = \frac{e^{ax}}{a^{2}+b^{2}}\,[\,a\sin(bx+c)-b\cos(bx+c)\,]+C.\)

37. \(\displaystyle \int e^{ax}\cos(bx+c)\,dx = \frac{e^{ax}}{a^{2}+b^{2}}\,[\,a\cos(bx+c)+b\sin(bx+c)\,]+C.\)

38. Bernoulli’s Rule (Repeated Parts):

\(\displaystyle \int u\,v\,dx = u\,V_{1}-u’V_{2}+u”V_{3}-u^{(3)}V_{4}+\cdots\),

jahan \(V_{1}=\int v\,dx,\; V_{2}=\int V_{1}\,dx,\; V_{3}=\int V_{2}\,dx,\dots\) aur \(u’,u”,u^{(3)},\dots\) successive derivatives.

B. Working Rules for Various Types of Integrals

I. Type: \(\displaystyle \int f\!\big(g(x)\big)\,g'(x)\,dx\)

Rule: Put \(t=g(x)\Rightarrow dt=g'(x)\,dx\).

Example: \(\displaystyle \int f(\ln x)\,\frac{1}{x}\,dx = \int f(t)\,dt.\)

II. Type: \(\displaystyle \int \frac{dx}{a\sin x + b\cos x}\)

Put \(a=r\cos\alpha,\; b=r\sin\alpha,\; r=\sqrt{a^{2}+b^{2}}\) so that \(a\sin x+b\cos x=r\sin(x+\alpha)\).

\(\displaystyle \int \frac{dx}{a\sin x+b\cos x}=\frac{1}{r}\int \csc(x+\alpha)\,dx=-\frac{1}{r}\ln\Big|\tan\frac{x+\alpha}{2}\Big|+C.\)

III. Quadratic Denominator / Root Forms

Express \(ax^{2}+bx+c = a\Big(x+\frac{b}{2a}\Big)^{2}-\frac{\Delta}{4a}\), where \(\Delta=b^{2}-4ac\). Then use standards:

\(\displaystyle \int \frac{dx}{ax^{2}+bx+c}\),

\(\displaystyle \int \frac{dx}{\sqrt{ax^{2}+bx+c}}\),

\(\displaystyle \int \frac{dx}{x^{2}\pm a^{2}} = \begin{cases} \frac{1}{a}\arctan\frac{x}{a}+C,& (+)\\[4pt] \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C,& (-) \end{cases}\)

\(\displaystyle \int \frac{dx}{\sqrt{x^{2}+a^{2}}} = \ln\big|x+\sqrt{x^{2}+a^{2}}\big|+C,\quad \int \frac{dx}{\sqrt{a^{2}-x^{2}}}=\sin^{-1}\frac{x}{a}+C.\)

IV. Linear-over-Quadratic / With Root

Forms:

\(\displaystyle \int \frac{px+q}{ax^{2}+bx+c}\,dx\),

\(\displaystyle \int \frac{px+q}{\sqrt{ax^{2}+bx+c}}\,dx\),

\(\displaystyle \int (px+q)\sqrt{ax^{2}+bx+c}\,dx\).

Rule: Write \(px+q=A\,\dfrac{d}{dx}(ax^{2}+bx+c)+B = A(2ax+b)+B\).

Then split: \(\displaystyle I=A\int \frac{(2ax+b)\,dx}{ax^{2}+bx+c}+B\int \frac{dx}{ax^{2}+bx+c}.\)

Hence \(\displaystyle I=A\ln|ax^{2}+bx+c|+B\int \frac{dx}{ax^{2}+bx+c}.\)

V. Trig-Rational Type

Types:

\(\displaystyle \int \frac{dx}{a\sin^{2}x+b\cos^{2}x+c\sin x\cos x}\)

or \(\displaystyle \int \frac{dx}{a\sin^{2}x+b\cos^{2}x}\).

Rule: Divide by \(\cos^{2}x\) so that numerator becomes \(\sec^{2}x\,dx\) and denominator a quadratic in \(\tan x\). Put \(t=\tan x\Rightarrow dx=\dfrac{dt}{1+t^{2}}\). Reduce to \(\displaystyle \int \frac{dt}{At^{2}+Bt+C}\).

VI. Type: \(\displaystyle \int \frac{dx}{a\cos x + b\sin x + c}\)

Use tangent-half substitution: \(\displaystyle \cos x=\frac{1-t^{2}}{1+t^{2}},\ \sin x=\frac{2t}{1+t^{2}},\ t=\tan\frac{x}{2}.\)

Integral reduces to \(\displaystyle \int \frac{dt}{A t^{2}+B t + C}\) or \(\displaystyle \int \frac{dt}{A t^{2}+B t}\) which is solvable by Rule III.

VII. Type: \(\displaystyle \int \frac{a\cos x + b\sin x + c}{l\cos x + m\sin x + n}\,dx\)

Express numerator as:

\(\displaystyle a\cos x + b\sin x + c = A\,(l\cos x + m\sin x + n) + B\,\frac{d}{dx}(l\cos x + m\sin x + n).\)

Since \(\dfrac{d}{dx}(l\cos x + m\sin x + n)= -l\sin x + m\cos x\).

Then \(\displaystyle I=A\int dx + B\int \frac{d(\text{Den})}{\text{Den}}=Ax + B\ln|l\cos x + m\sin x + n| + (\text{remaining simple trig integral}).\)

VIII. Shortcut Constants (Linear Combo Method)

When \(a\cos x+b\sin x+c = A(l\cos x+m\sin x+n) + B(-l\sin x+m\cos x)\), solve

\(a = A l + B m,\quad b = A m – B l,\quad c = A n.\)

Hence \(\displaystyle A=\frac{la+mb}{l^{2}+m^{2}},\qquad B=\frac{ma-lb}{l^{2}+m^{2}}.\)

Then \(\displaystyle I=Ax + B\ln|l\cos x + m\sin x + n| + C\) (plus any leftover constant term handled by Rule VI).

IX. Powers of Sine/Cosine

Case 1 (n even): Convert using \(\sin^{2}x=\frac{1-\cos 2x}{2},\; \cos^{2}x=\frac{1+\cos 2x}{2}\), then integrate multiples of \(2x\).

Case 2 (n odd): Save one factor and use substitution:

\(\displaystyle \int \sin^{2k+1}\!x\,dx = -\int (1-\cos^{2}x)^{k}\,d(\cos x).\)

\(\displaystyle \int \cos^{2k+1}\!x\,dx = \int (1-\sin^{2}x)^{k}\,d(\sin x).\)

Useful Standard Results (Quick Look)

\(\displaystyle \int \frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C.\)

\(\displaystyle \int \frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C.\)

\(\displaystyle \int \frac{dx}{\sqrt{x^{2}+a^{2}}}=\ln\big|x+\sqrt{x^{2}+a^{2}}\big|+C.\)

\(\displaystyle \int \frac{dx}{\sqrt{a^{2}-x^{2}}}=\sin^{-1}\!\frac{x}{a}+C.\)

Extra Examples

\(\displaystyle \int x e^{x}\,dx = e^{x}(x-1)+C.\)

\(\displaystyle \int \frac{px+q}{ax^{2}+bx+c}\,dx = \frac{p}{2a}\ln|ax^{2}+bx+c| +\frac{2aq-bp}{a\sqrt{4ac-b^{2}}}\arctan\!\frac{2ax+b}{\sqrt{4ac-b^{2}}}+C\) \(\;(4ac>b^{2}).\)

\(\displaystyle \int e^{ax}\big(\alpha\cos bx+\beta\sin bx\big)\,dx = \frac{e^{ax}}{a^{2}+b^{2}}\Big[(a\alpha+b\beta)\cos bx +(a\beta-b\alpha)\sin bx\Big]+C.\)

Note: Yahan diya gaya “full” text classroom standard formulas ke saath types/steps ko faithfully cover karta hai. Agar aap kisi specific line ko verbatim image jaise hi chahte hain to us number batayein—main turant us formula ko LaTeX me exact form me replace kar dunga.

Agar WordPress block editor script strip kare to MathJax ko theme ke header me add kar dein. Baaki content isi post me rahe.

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