Integration by Parts

Interactive Feed

JEE Main 2014

The integral $\int (1 + x - \frac{1}{x}) e^{x + 1/ x} d x$ is equal to

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

(B)

- x e^{x + 1/ x} + c

(C)

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

(D)

x e^{x + 1/ x} + c

JEE Main 2013

LEVELJEE Main

If $\int f (x) d x = ψ (x)$ , then $\int x^{5} f (x^{3}) d x$ is equal to

(A)

\frac{1}{3} [x^{3} ψ (x^{3}) - \int x^{2} ψ (x^{3}) d x] + C

(B)

\frac{1}{3} x^{3} ψ (x^{3}) - 3 \int x^{3} ψ (x^{3}) d x + C

(C)

\frac{1}{3} x^{3} ψ (x^{3}) - \int x^{2} ψ (x^{3}) d x + C

(D)

\frac{1}{3} [x^{3} ψ (x^{3}) - \int x^{3} ψ (x^{3}) d x] + C

JEE Advanced 1994

LEVELJEE Main

Find the indefinite integral $\int cos 2 θ ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) d θ$

JEE Advanced 1981

LEVELJEE Main

Evaluate $\int (e^{l o g x} + sin x) cos x d x$

JEE Main 2020 (4 September Shift 1)

LEVELJEE Main

The integral $\int (\frac{x}{x s i n x + c o s x})^{2} d x$ is equal to (where $C$ is a constant of integration) :

(A)

tan x - \frac{x s e c x}{x s i n x + c o s x} + C

(B)

sec x - \frac{x t a n x}{x s i n x + c o s x} + C

(C)

tan x + \frac{x s e c x}{x s i n x + c o s x} + C

(D)

sec x + \frac{x t a n x}{x s i n x + c o s x} + C

JEE Main 2023 (06 April Shift 1)

LEVELJEE Advanced

Let $I (x) = \int \frac{x ^{2} ( x s e c ^{2} x + t a n x )}{( x t a n x + 1 ) ^{2}} d x$ . If $I (0) = 0$ , then $I (\frac{π}{4})$ is equal to

(A)

lo g_{e} \frac{( π + 4 ) ^{2}}{16} + \frac{π ^{2}}{4 ( π + 4 )}

(B)

lo g_{e} \frac{( π + 4 ) ^{2}}{16} - \frac{π ^{2}}{4 ( π + 4 )}

(C)

lo g_{e} \frac{( π + 4 ) ^{2}}{32} - \frac{π ^{2}}{4 ( π + 4 )}

(D)

lo g_{e} \frac{( π + 4 ) ^{2}}{32} + \frac{π ^{2}}{4 ( π + 4 )}

JEE Main 2025 (January)

LEVELJEE Main

Let $\int x^{3} sin x d x = g (x) + C$ where C is the constant of integration. If $8 (g (\frac{π}{2}) + g^{'} (\frac{π}{2})) = α π^{3} + β π^{2} + γ .$ $α, β, γ \in Z$ , then $α + β - γ$ equals :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫(1+x−x1​)ex+1/xdx is equal to

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the indefinite integral ∫cos2θln(cosθ−sinθcosθ+sinθ​)dθ

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫(xsinx+cosxx​)2dx 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 I(x)=∫(xtanx+1)2x2(xsec2x+tanx)​dx. If I(0)=0, then I(4π​) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ∫x3sin x dx=g(x)+C where C is the constant of integration. If 8(g(2π​)+g′(2π​))=απ3+βπ2+γ. α,β,γ∈Z, then α+β−γ equals :

The integral $\int (1 + x - \frac{1}{x}) e^{x + 1/ x} d x$ is equal to

If $\int f (x) d x = ψ (x)$ , then $\int x^{5} f (x^{3}) d x$ is equal to

Find the indefinite integral $\int cos 2 θ ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) d θ$

Evaluate $\int (e^{l o g x} + sin x) cos x d x$

The integral $\int (\frac{x}{x s i n x + c o s x})^{2} d x$ is equal to (where $C$ is a constant of integration) :

Let $I (x) = \int \frac{x ^{2} ( x s e c ^{2} x + t a n x )}{( x t a n x + 1 ) ^{2}} d x$ . If $I (0) = 0$ , then $I (\frac{π}{4})$ is equal to

Let $\int x^{3} sin x d x = g (x) + C$ where C is the constant of integration. If $8 (g (\frac{π}{2}) + g^{'} (\frac{π}{2})) = α π^{3} + β π^{2} + γ .$ $α, β, γ \in Z$ , then $α + β - γ$ equals :