Phrases Indefinite Integration Previous Year Question Paper and Solution with PDF

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Phrases Indefinite Integration PYQ

Qus : 1

$\int f(x)\mathrm{d}x=g(x)$ , then

$\int {x}^5f({x}^3)\mathrm{d}x$

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Solution

Quick Solution

Given:

$\int f(x)\, dx = g(x)$

Required: $\int x^5 f(x^3)\, dx$

Use substitution:

Let $u = x^3 \Rightarrow du = 3x^2\, dx \Rightarrow dx = \frac{du}{3x^2}$

Now rewrite the integral:

$\int x^5 f(x^3)\, dx = \int x^5 f(u) \cdot \frac{du}{3x^2} = \frac{1}{3} \int x^3 f(u)\, du$

But $x^3 = u$ , so:

$\frac{1}{3} \int u f(u)\, du$

Now integrate by parts or use the identity:

$\int u f(u)\, du = u g(u) - \int g(u)\, du$

Final answer:

$\int x^5 f(x^3)\, dx = \frac{1}{3} \left[ x^3 g(x^3) - \int g(x^3) \cdot 3x^2\, dx \right] = x^3 g(x^3) - \int x^2 g(x^3)\, dx$

$\boxed{ \int x^5 f(x^3)\, dx = x^3 g(x^3) - \int x^2 g(x^3)\, dx }$

Qus : 2

$\int x\, \sin x\, sec^3x\, dx=\frac{1}{2}\Bigg{[}f(x){se}c^2x+g(x)\Bigg{(}\frac{\tan x}{x}\Bigg{)}\Bigg{]}+C$ , then which of the following is true?

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Solution

Qus : 3

The integral

$\int \sqrt{1+2 cot x(cosec x+cotx)} dx$ ,

$(0<x<\frac{\pi}{2})$ (where C is a constant of integration) is equal to

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Solution

Qus : 4

, then the values of A₁, A₂, A₃, A₄ are

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Solution

Qus : 5

The value of

$\int \frac{({x}^2-1)}{{x}^3\sqrt[]{2{x}^4-2{x}^2+1}}dx$

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Solution

Qus : 6

The value of

$\int \sqrt{x} e^{\sqrt{x}} dx$ is equal to:

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Solution

Qus : 7

$\int {3}^{{3}^{{3}^x}}.{3}^{{3}^x}.{3}^xdx$ is equal to

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Solution

Qus : 8

The value of

$\int \frac{(x+1)}{x(xe^{x}+1)} dx$ is equal to

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Solution

Qus : 9

$\int e^{x}(f(x)-f'(x))dx=\phi(x)$ , then the value of

$\int e^x f(x) dx$ is

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Solution

Qus : 10

then value of k is

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Solution

Qus : 11

Evaluate

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Solution

Qus : 12

$\int \frac{xe^{x}}{\sqrt{1+e^{x}}}=f(x)\sqrt{1+e^{x}}-2log \frac{\sqrt{1+e^{x}}-1}{\sqrt{1+e^{x}}+1}+C$ then

$f(x)$ is

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Solution

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