Integration by Substitution

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JEE Main 2016

The integral $\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x$ is equal to

\frac{x ^{5}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

(B)

\frac{- x ^{10}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

(C)

\frac{- x ^{5}}{( x ^{5} + x ^{3} + 1 ) ^{2}} + C

(D)

\frac{x ^{10}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

JEE Main 2015

LEVELJEE Main

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

(A)

- (x^{4} + 1)^{1/4} + c

(B)

- (\frac{x ^{4} + 1}{x ^{4}})^{1/4} + c

(C)

(\frac{x ^{4} + 1}{x ^{4}})^{1/4} + c

(D)

(x^{4} + 1)^{1/4} + c

JEE Main 2008

LEVELBoard

The value of $2 \int \frac{s i n x d x}{s i n ( x - \frac{π}{4} )}$ is

(A)

x + lo g cos (x - \frac{π}{4}) + c

(B)

x - lo g sin (x - \frac{π}{4}) + c

(C)

x + lo g sin (x - \frac{π}{4}) + c

(D)

x - lo g cos (x - \frac{π}{4}) + c

JEE Main 2007

LEVELJEE Main

$\int \frac{d x}{c o s x + 3 s i n x}$ equals

(A)

lo g tan (\frac{x}{2} + \frac{π}{12}) + C

(B)

lo g tan (\frac{x}{2} - \frac{π}{12}) + C

(C)

\frac{1}{2} lo g tan (\frac{x}{2} + \frac{π}{12}) + C

(D)

\frac{1}{2} lo g tan (\frac{x}{2} - \frac{π}{12}) + C

JEE Advanced 2012

LEVELJEE Main

The integral $\int \frac{s e c ^{2} x}{( s e c x + t a n x ) ^{9/2}} d x$ equals (for some arbitrary constant $K$ )

(A)

\frac{1}{( s e c x + t a n x ) ^{11/2}} {\frac{1}{11} - \frac{1}{7} (sec x + tan x)^{2}} + K

(B)

\frac{1}{( s e c x + t a n x ) ^{11/2}} {\frac{1}{11} - \frac{1}{7} (sec x + tan x)^{2}} + K

(C)

\frac{1}{( s e c x + t a n x ) ^{11/2}} {\frac{1}{11} + \frac{1}{7} (sec x + tan x)^{2}} + K

(D)

\frac{1}{( s e c x + t a n x ) ^{11/2}} {\frac{1}{11} + \frac{1}{7} (sec x + tan x)^{2}} + K

JEE Advanced 2007

LEVELJEE Main

Let $f (x) = \frac{x}{( 1 + x ^{n} ) ^{1/ n}}$ for $n \geq 2$ and $g (x) = (f \circ f \circ ... \circ f) (x)$ ( $f$ occurs $n$ times). Then $\int x^{n - 2} g (x) d x$ equals

(A)

\frac{1}{n ( n - 1 )} (1 + n x^{n})^{1 - 1/ n} + K

(B)

\frac{1}{n - 1} (1 + n x^{n})^{1 - 1/ n} + K

(C)

\frac{1}{n ( n + 1 )} (1 + n x^{n})^{1 + 1/ n} + K

(D)

\frac{1}{n + 1} (1 + n x^{n})^{1 + 1/ n} + K

JEE Advanced 2006

LEVELJEE Main

$\int \frac{x ^{2} - 1}{x ^{3} 2 x ^{4} - 2 x ^{2} + 1} d x =$

(A)

\frac{2 x ^{4} - 2 x ^{2} + 1}{x ^{2}} + c

(B)

\frac{2 x ^{4} - 2 x ^{2} + 1}{x ^{3}} + c

(C)

\frac{2 x ^{4} - 2 x ^{2} + 1}{x} + c

(D)

\frac{2 x ^{4} - 2 x ^{2} + 1}{2 x ^{2}} + c

JEE Advanced 1995

LEVELJEE Main

The value of the integral $\int \frac{c o s ^{3} x + c o s ^{5} x}{s i n ^{2} x + s i n ^{4} x} d x$ is

(A)

sin x - 6 tan^{- 1} (sin x) + c

(B)

sin x - 2 (sin x)^{- 1} + c

(C)

sin x - 2 (sin x)^{- 1} - 6 tan^{- 1} (sin x) + c

(D)

sin x - 2 (sin x)^{- 1} + 5 tan^{- 1} (sin x) + c

JEE Advanced 2002

LEVELJEE Main

For any natural number $m$ , evaluate $\int (x^{3 m} + x^{2 m} + x^{m}) (2 x^{2 m} + 3 x^{m} + 6)^{1/ m} d x, x > 0$

JEE Advanced 2001

LEVELJEE Main

Evaluate $\int sin^{- 1} (\frac{2 x + 2}{4 x ^{2} + 8 x + 13}) d x$

JEE Advanced 1996

LEVELJEE Main

Evaluate $\int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x$

JEE Advanced 1992

LEVELJEE Advanced

Find the indefinite integral $\int (\frac{1}{3 x + 4 x} + \frac{l n ( 1 + 6 x )}{3 x + x}) d x$

JEE Advanced 1989

LEVELJEE Main

The integral $\int \frac{s i n ^{2} x c o s ^{2} x}{( s i n ^{5} x + c o s ^{3} x s i n ^{2} x + s i n ^{3} x c o s ^{2} x + c o s ^{5} x ) ^{2}} d x$ is equal to :

(A)

\frac{- 1}{1 + c o t ^{3} x} + C

(B)

\frac{1}{3 ( 1 + t a n ^{3} x )} + C

(C)

\frac{- 1}{3 ( 1 + t a n ^{3} x )} + C

(D)

\frac{1}{1 + c o t ^{3} x} + C

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

If $f (\frac{x - 4}{x + 2}) = 2 x + 1, (x \in R - {1, - 2})$ , then $\int f (x) d x$ is equal to : (where C is a constant of integration)

(A)

12 lo g_{e} ∣1 - x ∣ - 3 x + c

(B)

- 12 lo g_{e} ∣1 - x ∣ - 3 x + c

(C)

- 12 lo g_{e} ∣1 - x ∣ + 3 x + c

(D)

12 lo g_{e} ∣1 - x ∣ + 3 x + c

JEE Main 2019 (9 January)

LEVELJEE Main

For $x^{2} \neq = nπ + 1$ , $n \in N$ (the set of natural numbers), the integral $\int \frac{2 s i n ( x ^{2} - 1 ) - s i n 2 ( x ^{2} - 1 )}{2 s i n ( x ^{2} - 1 ) + s i n 2 ( x ^{2} - 1 )} d x$ is equal to : (where c is a constant of integration)

(A)

\frac{1}{2} lo g_{e} ∣ sec (\frac{x ^{2} - 1}{2}) ∣ + c

(B)

lo g_{e} ∣ sec^{2} (x^{2} - 1) ∣ + c

(C)

\frac{1}{2} lo g_{e} ∣ sec^{2} (\frac{x ^{2} - 1}{2}) ∣ + c

(D)

\frac{1}{2} lo g_{e} ∣ sec (x^{2} - 1) ∣ + c

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

Let $n \geq 2$ be a natural number and $0 < θ < π /2$ . Then $\int \frac{( s i n ^{n} θ - s i n θ ) ^{1/ n} c o s θ}{s i n ^{n + 1} θ} d θ$ is equal to : (Where C is a constant of integration)

(A)

\frac{n}{n ^{2} - 1} (1 - \frac{1}{s i n ^{n - 1} θ})^{\frac{n + 1}{n}} + C

(B)

\frac{n}{n ^{2} + 1} (1 - \frac{1}{s i n ^{n - 1} θ})^{\frac{n + 1}{n}} + C

(C)

\frac{n}{n ^{2} - 1} (1 - \frac{1}{s i n ^{n - 1} θ})^{\frac{n + 1}{n}} + C

(D)

\frac{n}{n ^{2} - 1} (1 + \frac{1}{s i n ^{n - 1} θ})^{\frac{n + 1}{n}} + C

JEE Main 2019 (9 January)

LEVELJEE Main

If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0)$ and $f (0) = 0$ , then the value of $f (1)$ is :

JEE Main 2019 (12 January Shift 1)

LEVELBoard

The integral $\int cos (lo g_{e} x) d x$ is equal to : (where C is a constant of integration)

(A)

\frac{x}{2} [sin (lo g_{e} x) - cos (lo g_{e} x)] + C

(B)

\frac{x}{2} [cos (lo g_{e} x) + sin (lo g_{e} x)] + C

(C)

x [cos (lo g_{e} x) + sin (lo g_{e} x)] + C

(D)

x [cos (lo g_{e} x) - sin (lo g_{e} x)] + C

JEE Main 2019 (11 January)

LEVELJEE Main

If $\int \frac{1 - x ^{2}}{x ^{4}} d x = A (x) (1 - x^{2})^{m} + C$ , for a suitable chosen integer $m$ and a function $A (x)$ , where $C$ is a constant of integration then $(A (x))^{m}$ equals :

(A)

\frac{- 1}{3 x ^{3}}

(B)

\frac{- 1}{27 x ^{9}}

(C)

\frac{1}{9 x ^{4}}

(D)

\frac{1}{27 x ^{6}}

JEE Main 2019 (08 April Shift 2)

LEVELJEE Main

If $\int \frac{d x}{x ^{3} ( 1 + x ^{6} ) ^{2/3}} = x f (x) (1 + x^{6})^{1/3} + C$ where C is a constant of integration, then the function f(x) is equal to-

(A)

1/ (6 x^{3})

(B)

3/ x^{2}

(C)

1/ (2 x^{2})

(D)

1/ (2 x^{3})

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫(secx+tanx)9/2sec2x​dx equals (for some arbitrary constant K)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=(1+xn)1/nx​ for n≥2 and g(x)=(f∘f∘...∘f)(x) (f occurs n times). Then ∫xn−2g(x)dx equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∫x32x4−2x2+1​x2−1​dx=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of the integral ∫sin2x+sin4xcos3x+cos5x​dx is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any natural number m, evaluate ∫(x3m+x2m+xm)(2x2m+3xm+6)1/mdx,x>0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫sin−1(4x2+8x+13​2x+2​)dx

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫x(1+xex)2(x+1)​dx

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the indefinite integral ∫(3x​+4x​1​+3x​+x​ln(1+6x​)​)dx

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫(tanx​+cotx​)dx

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫sinx(cos2x)1/2​dx

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate the following ∫1+x​1−x​​​dx

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate the following ∫x2(x4+1)3/4dx​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫(a+bx)2x2dx​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫(sin5x+cos3xsin2x+sin3xcos2x+cos5x)2sin2xcos2x​dx is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x+2x−4​)=2x+1,(x∈R−{1,−2}), then ∫f(x)dx is equal to : (where C is a constant of integration)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let n≥2 be a natural number and 0<θ<π/2. Then ∫sinn+1θ(sinnθ−sinθ)1/ncosθ​dθ is equal to : (Where C is a constant of integration)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=∫(x2+1+2x7)25x8+7x6​dx,(x≥0) and f(0)=0, then the value of f(1) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫cos(loge​x)dx is equal to : (where C is a constant of integration)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∫x41−x2​​dx=A(x)(1−x2​)m+C, for a suitable chosen integer m and a function A(x), where C is a constant of integration then (A(x))m equals :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∫x3(1+x6)2/3dx​=xf(x)(1+x6)1/3+C where C is a constant of integration, then the function f(x) is equal to-

The integral $\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x$ is equal to

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

The value of $2 \int \frac{s i n x d x}{s i n ( x - \frac{π}{4} )}$ is

$\int \frac{d x}{c o s x + 3 s i n x}$ equals

The integral $\int \frac{s e c ^{2} x}{( s e c x + t a n x ) ^{9/2}} d x$ equals (for some arbitrary constant $K$ )

Let $f (x) = \frac{x}{( 1 + x ^{n} ) ^{1/ n}}$ for $n \geq 2$ and $g (x) = (f \circ f \circ ... \circ f) (x)$ ( $f$ occurs $n$ times). Then $\int x^{n - 2} g (x) d x$ equals

$\int \frac{x ^{2} - 1}{x ^{3} 2 x ^{4} - 2 x ^{2} + 1} d x =$

The value of the integral $\int \frac{c o s ^{3} x + c o s ^{5} x}{s i n ^{2} x + s i n ^{4} x} d x$ is

For any natural number $m$ , evaluate $\int (x^{3 m} + x^{2 m} + x^{m}) (2 x^{2 m} + 3 x^{m} + 6)^{1/ m} d x, x > 0$

Evaluate $\int sin^{- 1} (\frac{2 x + 2}{4 x ^{2} + 8 x + 13}) d x$

Evaluate $\int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x$

Find the indefinite integral $\int (\frac{1}{3 x + 4 x} + \frac{l n ( 1 + 6 x )}{3 x + x}) d x$

Evaluate $\int (tan x + cot x) d x$

Evaluate $\int \frac{( c o s 2 x ) ^{1/2}}{s i n x} d x$

Evaluate the following $\int \frac{1 - x}{1 + x} d x$

Evaluate the following $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$

Evaluate $\int \frac{x ^{2} d x}{( a + b x ) ^{2}}$

The integral $\int \frac{s i n ^{2} x c o s ^{2} x}{( s i n ^{5} x + c o s ^{3} x s i n ^{2} x + s i n ^{3} x c o s ^{2} x + c o s ^{5} x ) ^{2}} d x$ is equal to :

If $f (\frac{x - 4}{x + 2}) = 2 x + 1, (x \in R - {1, - 2})$ , then $\int f (x) d x$ is equal to : (where C is a constant of integration)

Let $n \geq 2$ be a natural number and $0 < θ < π /2$ . Then $\int \frac{( s i n ^{n} θ - s i n θ ) ^{1/ n} c o s θ}{s i n ^{n + 1} θ} d θ$ is equal to : (Where C is a constant of integration)

If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0)$ and $f (0) = 0$ , then the value of $f (1)$ is :

The integral $\int cos (lo g_{e} x) d x$ is equal to : (where C is a constant of integration)

If $\int \frac{1 - x ^{2}}{x ^{4}} d x = A (x) (1 - x^{2})^{m} + C$ , for a suitable chosen integer $m$ and a function $A (x)$ , where $C$ is a constant of integration then $(A (x))^{m}$ equals :

If $\int \frac{d x}{x ^{3} ( 1 + x ^{6} ) ^{2/3}} = x f (x) (1 + x^{6})^{1/3} + C$ where C is a constant of integration, then the function f(x) is equal to-