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Indefinite Integration formulas

Master Indefinite Integration through 34 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Indefinite Integration, every formula

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Power rule
$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$
Q1MCQPower rule
$\displaystyle\int x^{3}\,dx$ equals:
- A
  $\dfrac{x^{4}}{4}+C$
- B
  $3x^{2}+C$
- C
  $\dfrac{x^{4}}{3}+C$
- D
  $x^{4}+C$
Reciprocal
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
Q1MCQReciprocal
$\displaystyle\int\dfrac{1}{x}\,dx$ equals:
- A
  $\ln|x|+C$
- B
  $-\dfrac{1}{x^{2}}+C$
- C
  $x^{-1}+C$
- D
  $\ln x^{2}+C$
Exponential
$\int e^{x} d x = e^{x} + C$
Q1MCQExponential
$\displaystyle\int e^{x}\,dx$ equals:
- A
  $e^{x}+C$
- B
  $xe^{x-1}+C$
- C
  $\dfrac{e^{x}}{x}+C$
- D
  $e^{x}\ln x+C$
General exponential
$\int a^{x} d x = \frac{a ^{x}}{ln a} + C$
a>0, a≠1
Q1MCQGeneral exponential
$\displaystyle\int 2^{x}\,dx$ equals:
- A
  $\dfrac{2^{x}}{\ln 2}+C$
- B
  $2^{x}\ln 2+C$
- C
  $x\,2^{x-1}+C$
- D
  $\dfrac{2^{x+1}}{x+1}+C$
Sine
$\int sin x d x = - cos x + C$
Q1MCQSine
$\displaystyle\int\sin x\,dx$ equals:
- A
  $-\cos x+C$
- B
  $\cos x+C$
- C
  $-\sin x+C$
- D
  $\sin x+C$
Cosine
$\int cos x d x = sin x + C$
Q1MCQCosine
$\displaystyle\int\cos x\,dx$ equals:
- A
  $\sin x+C$
- B
  $-\sin x+C$
- C
  $\cos x+C$
- D
  $-\cos x+C$
sec^2
$\int sec^{2} x d x = tan x + C$
Q1MCQsec^2
$\displaystyle\int\sec^{2}x\,dx$ equals:
- A
  $\tan x+C$
- B
  $-\cot x+C$
- C
  $\sec x+C$
- D
  $\tan^{2}x+C$
cosec^2
$\int csc^{2} x d x = - cot x + C$
Q1MCQcosec^2
$\displaystyle\int\csc^{2}x\,dx$ equals:
- A
  $-\cot x+C$
- B
  $\cot x+C$
- C
  $\tan x+C$
- D
  $-\csc x+C$
sec·tan
$\int sec x tan x d x = sec x + C$
Q1MCQsec·tan
$\displaystyle\int\sec x\tan x\,dx$ equals:
- A
  $\sec x+C$
- B
  $\tan x+C$
- C
  $\sec^{2}x+C$
- D
  $-\sec x+C$
Integral of tan
$\int tan x d x = ln ∣ sec x ∣ + C$
Q1MCQIntegral of tan
$\displaystyle\int\tan x\,dx$ equals:
- A
  $\ln|\sec x|+C$
- B
  $\sec^{2}x+C$
- C
  $\ln|\cos x|+C$
- D
  $-\ln|\sec x|+C$
Integral of sec
$\int sec x d x = ln ∣ sec x + tan x ∣ + C$
Q1MCQIntegral of sec
$\displaystyle\int\sec x\,dx$ equals:
- A
  $\ln|\sec x+\tan x|+C$
- B
  $\sec x\tan x+C$
- C
  $\tan x+C$
- D
  $\ln|\sec x|+C$
arcsin form
$\int \frac{d x}{1 - x ^{2}} = sin^{- 1} x + C$
Q1MCQarcsin form
$\displaystyle\int\dfrac{dx}{\sqrt{1-x^{2}}}$ equals:
- A
  $\sin^{-1}x+C$
- B
  $\tan^{-1}x+C$
- C
  $\cos^{-1}x+C$
- D
  $\sec^{-1}x+C$
arctan form
$\int \frac{d x}{1 + x ^{2}} = tan^{- 1} x + C$
Q1MCQarctan form
$\displaystyle\int\dfrac{dx}{1+x^{2}}$ equals:
- A
  $\tan^{-1}x+C$
- B
  $\sin^{-1}x+C$
- C
  $\ln(1+x^{2})+C$
- D
  $\cot^{-1}x+C$
Linear substitution
$\int f (a x + b) d x = \frac{F ( a x + b )}{a} + C$
F'=f
Q1MCQLinear substitution
$\displaystyle\int\cos(3x)\,dx$ equals:
- A
  $\dfrac{\sin 3x}{3}+C$
- B
  $\sin 3x+C$
- C
  $3\sin 3x+C$
- D
  $-\dfrac{\sin 3x}{3}+C$
Log-derivative rule
$\int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣ + C$
Q1MCQLog-derivative
$\displaystyle\int\dfrac{2x}{x^{2}+1}\,dx$ equals:
- A
  $\ln(x^{2}+1)+C$
- B
  $\tan^{-1}x+C$
- C
  $\dfrac{1}{x^{2}+1}+C$
- D
  $2\ln x+C$
Power of a function
$\int [f (x)]^{n} f^{'} (x) d x = \frac{[ f ( x ) ] ^{n + 1}}{n + 1} + C, n \neq = - 1$
Q1MCQPower of a function
$\displaystyle\int (x^{2}+1)^{3}\,(2x)\,dx$ equals:
- A
  $\dfrac{(x^{2}+1)^{4}}{4}+C$
- B
  $(x^{2}+1)^{4}+C$
- C
  $\dfrac{(x^{2}+1)^{3}}{3}+C$
- D
  $3(x^{2}+1)^{2}+C$
Root of a function
$\int \frac{f ^{'} ( x )}{f ( x )} d x = 2 f (x) + C$
Q1MCQRoot of a function
$\displaystyle\int\dfrac{2x}{\sqrt{x^{2}+1}}\,dx$ equals:
- A
  $2\sqrt{x^{2}+1}+C$
- B
  $\sqrt{x^{2}+1}+C$
- C
  $\ln\sqrt{x^{2}+1}+C$
- D
  $\dfrac{1}{\sqrt{x^{2}+1}}+C$
arcsin (a)
$\int \frac{d x}{a ^{2} - x ^{2}} = sin^{- 1} \frac{x}{a} + C$
Q1MCQarcsin (a)
$\displaystyle\int\dfrac{dx}{\sqrt{9-x^{2}}}$ equals:
- A
  $\sin^{-1}\dfrac{x}{3}+C$
- B
  $\dfrac13\sin^{-1}\dfrac{x}{3}+C$
- C
  $\sin^{-1}\dfrac{x}{9}+C$
- D
  $\tan^{-1}\dfrac{x}{3}+C$
arctan (a)
$\int \frac{d x}{a ^{2} + x ^{2}} = \frac{1}{a} tan^{- 1} \frac{x}{a} + C$
Q1MCQarctan (a)
$\displaystyle\int\dfrac{dx}{4+x^{2}}$ equals:
- A
  $\dfrac12\tan^{-1}\dfrac{x}{2}+C$
- B
  $\tan^{-1}\dfrac{x}{2}+C$
- C
  $\dfrac14\tan^{-1}\dfrac{x}{2}+C$
- D
  $\dfrac12\sin^{-1}\dfrac{x}{2}+C$
1/(x^2-a^2)
$\int \frac{d x}{x ^{2} - a ^{2}} = \frac{1}{2 a} ln \frac{x - a}{x + a} + C$
Q1MCQ1/(x^2-a^2)
$\displaystyle\int\dfrac{dx}{x^{2}-9}$ equals:
- A
  $\dfrac{1}{6}\ln\left|\dfrac{x-3}{x+3}\right|+C$
- B
  $\dfrac{1}{3}\ln\left|\dfrac{x-3}{x+3}\right|+C$
- C
  $\dfrac{1}{6}\ln\left|\dfrac{x+3}{x-3}\right|+C$
- D
  $\tan^{-1}\dfrac{x}{3}+C$
1/(a^2-x^2)
$\int \frac{d x}{a ^{2} - x ^{2}} = \frac{1}{2 a} ln \frac{a + x}{a - x} + C$
Q1MCQ1/sqrt(x^2+a^2)
$\displaystyle\int\dfrac{dx}{\sqrt{x^{2}+4}}$ equals:
- A
  $\ln\left|x+\sqrt{x^{2}+4}\right|+C$
- B
  $\sin^{-1}\dfrac{x}{2}+C$
- C
  $\dfrac12\ln\left|x+\sqrt{x^{2}+4}\right|+C$
- D
  $\tan^{-1}\dfrac{x}{2}+C$
1/sqrt(x^2+a^2)
$\int \frac{d x}{x ^{2} + a ^{2}} = ln x + x^{2} + a^{2} + C$
Q1MCQ1/sqrt(x^2-a^2)
$\displaystyle\int\dfrac{dx}{\sqrt{x^{2}-1}}$ equals:
- A
  $\ln\left|x+\sqrt{x^{2}-1}\right|+C$
- B
  $\sin^{-1}x+C$
- C
  $\sec^{-1}x+C$
- D
  $\ln|x-1|+C$
1/sqrt(x^2-a^2)
$\int \frac{d x}{x ^{2} - a ^{2}} = ln x + x^{2} - a^{2} + C$
Q1MCQ1/(a^2-x^2)
$\displaystyle\int\dfrac{dx}{1-x^{2}}$ equals:
- A
  $\dfrac12\ln\left|\dfrac{1+x}{1-x}\right|+C$
- B
  $\tan^{-1}x+C$
- C
  $\sin^{-1}x+C$
- D
  $\dfrac12\ln\left|\dfrac{1-x}{1+x}\right|+C$
sqrt(a^2-x^2)
$\int a^{2} - x^{2} d x = \frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a} + C$
Q1MCQsqrt(a^2-x^2)
$\displaystyle\int\sqrt{a^{2}-x^{2}}\,dx$ equals:
- A
  $\dfrac{x}{2}\sqrt{a^{2}-x^{2}}+\dfrac{a^{2}}{2}\sin^{-1}\dfrac{x}{a}+C$
- B
  $\dfrac{(a^{2}-x^{2})^{3/2}}{3}+C$
- C
  $\dfrac{x}{2}\sqrt{a^{2}-x^{2}}+C$
- D
  $a^{2}\sin^{-1}\dfrac{x}{a}+C$
sqrt(x^2+a^2)
$\int x^{2} + a^{2} d x = \frac{x}{2} x^{2} + a^{2} + \frac{a ^{2}}{2} ln x + x^{2} + a^{2} + C$
Q1MCQsqrt(x^2+a^2)
The formula for $\displaystyle\int\sqrt{x^{2}+a^{2}}\,dx$ has leading term:
- A
  $\dfrac{x}{2}\sqrt{x^{2}+a^{2}}$
- B
  $x\sqrt{x^{2}+a^{2}}$
- C
  $\dfrac{(x^{2}+a^{2})^{3/2}}{3}$
- D
  $\dfrac{a^{2}}{2}\ln x$
Integration by parts
$\int u v d x = u \int v d x - \int (\frac{d u}{d x} \int v d x) d x$
first fn by ILATE
Q1MCQBy parts (ILATE)
In $\displaystyle\int x\,e^{x}\,dx$, by ILATE the first function is:
- A
  $x$
- B
  $e^{x}$
- C
  $xe^{x}$
- D
  either
e^x [f+f']
$\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x) + C$
Q1MCQe^x[f+f']
$\displaystyle\int e^{x}\big(\sin x+\cos x\big)\,dx$ equals:
- A
  $e^{x}\sin x+C$
- B
  $e^{x}\cos x+C$
- C
  $e^{x}(\sin x-\cos x)+C$
- D
  $-e^{x}\cos x+C$
e^x·(sin/cos)
$\int e^{a x} sin b x d x = \frac{e ^{a x}}{a ^{2} + b ^{2}} (a sin b x - b cos b x) + C$
Q1MCQe^x sin
$\displaystyle\int e^{x}\sin x\,dx$ equals:
- A
  $\dfrac{e^{x}}{2}(\sin x-\cos x)+C$
- B
  $\dfrac{e^{x}}{2}(\sin x+\cos x)+C$
- C
  $e^{x}\sin x+C$
- D
  $-e^{x}\cos x+C$
x^n·ln x
$\int x^{n} ln x d x = \frac{x ^{n + 1}}{n + 1} (ln x - \frac{1}{n + 1}) + C$
n≠-1
Q1MCQx^n ln x
$\displaystyle\int x\ln x\,dx$ equals:
- A
  $\dfrac{x^{2}}{2}\left(\ln x-\dfrac12\right)+C$
- B
  $\dfrac{x^{2}}{2}\ln x+C$
- C
  $x^{2}\ln x+C$
- D
  $\dfrac{x^{2}}{2}\left(\ln x+\dfrac12\right)+C$
Integral of arcsin
$\int sin^{- 1} x d x = x sin^{- 1} x + 1 - x^{2} + C$
Q1MCQIntegral of arcsin
$\displaystyle\int\sin^{-1}x\,dx$ equals:
- A
  $x\sin^{-1}x+\sqrt{1-x^{2}}+C$
- B
  $x\sin^{-1}x-\sqrt{1-x^{2}}+C$
- C
  $\sin^{-1}x+C$
- D
  $\dfrac{(\sin^{-1}x)^{2}}{2}+C$
Integral of arctan
$\int tan^{- 1} x d x = x tan^{- 1} x - \frac{1}{2} ln (1 + x^{2}) + C$
Q1MCQIntegral of arctan
$\displaystyle\int\tan^{-1}x\,dx$ equals:
- A
  $x\tan^{-1}x-\dfrac12\ln(1+x^{2})+C$
- B
  $x\tan^{-1}x+\dfrac12\ln(1+x^{2})+C$
- C
  $\tan^{-1}x+C$
- D
  $\dfrac{(\tan^{-1}x)^{2}}{2}+C$
Reduction: sin^n
$I_{n} = \frac{- sin ^{n - 1} x cos x}{n} + \frac{n - 1}{n} I_{n - 2}$
I_n=∫sin^n x dx
Q1MCQReduction sin^n
The reduction formula for $I_n=\int\sin^{n}x\,dx$ is:
- A
  $\dfrac{-\sin^{n-1}x\cos x}{n}+\dfrac{n-1}{n}I_{n-2}$
- B
  $\dfrac{\sin^{n-1}x\cos x}{n}+I_{n-2}$
- C
  $\dfrac{\cos^{n-1}x\sin x}{n}+\dfrac{n-1}{n}I_{n-2}$
- D
  $\dfrac{\sin^{n+1}x}{n+1}$
Reduction: tan^n
$I_{n} = \frac{tan ^{n - 1} x}{n - 1} - I_{n - 2}$
I_n=∫tan^n x dx
Q1MCQReduction tan^n
The reduction formula for $I_n=\int\tan^{n}x\,dx$ is:
- A
  $\dfrac{\tan^{n-1}x}{n-1}-I_{n-2}$
- B
  $\dfrac{\tan^{n-1}x}{n-1}+I_{n-2}$
- C
  $\dfrac{\tan^{n+1}x}{n+1}$
- D
  $\dfrac{\sec^{n-1}x}{n-1}-I_{n-2}$
Weierstrass substitution
$t = tan \frac{x}{2} : sin x = \frac{2 t}{1 + t ^{2}}, cos x = \frac{1 - t ^{2}}{1 + t ^{2}}, d x = \frac{2 d t}{1 + t ^{2}}$
for ∫dx/(a+b sin x) etc.
Q1MCQWeierstrass substitution
For $\displaystyle\int\dfrac{dx}{a+b\sin x}$, the substitution $t=\tan\tfrac{x}{2}$ gives $\sin x=$:
- A
  $\dfrac{2t}{1+t^{2}}$
- B
  $\dfrac{1-t^{2}}{1+t^{2}}$
- C
  $\dfrac{2t}{1-t^{2}}$
- D
  $\dfrac{1+t^{2}}{2t}$

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Indefinite Integration formulas

Indefinite Integration, every formula

Power rule

Reciprocal

Exponential

General exponential

Sine

Cosine

sec^2

cosec^2

sec·tan

Integral of tan

Integral of sec

arcsin form

arctan form

Linear substitution

Log-derivative rule

Power of a function

Root of a function

arcsin (a)

arctan (a)

1/(x^2-a^2)

1/(a^2-x^2)

1/sqrt(x^2+a^2)

1/sqrt(x^2-a^2)

sqrt(a^2-x^2)

sqrt(x^2+a^2)

Integration by parts

e^x [f+f']

e^x·(sin/cos)

x^n·ln x

Integral of arcsin

Integral of arctan

Reduction: sin^n

Reduction: tan^n

Weierstrass substitution

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