Indefinite Integration - Maharashtra Board STD 12 Science Maths Questions - Vidyadip

Question types

Indefinite Integration question types

43 questions across 5 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.

43

Questions

5

Question groups

5

Question types

MCQ

Answer the following questions in short.

Solve the Following Question.(2 Marks)

Solve the Following Question.(3 Marks)

Solve the Following Question.(4 Marks)

Sample Questions

Indefinite Integration questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1MCQ2 Marks

$\int \frac{\cos (2 x)-1}{\cos (2 x)+1} d x=$______.

A
tan x-x + c
B
x + tan x + c
C
x- tan x + c
D
- x - cot x + c

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Q 2MCQ2 Marks

$\int \frac{\cos (2 x)-1}{\cos (2 x)+1} d x=$ _____.

A
$\tan x-x+c$
B
$x+\tan x+c$
C
$x-\tan x+c$
D
$-x-\cot x+c$

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Q 3MCQ2 Marks

$\int \frac{1}{1+\cos x} d x=$______.

A
$\tan \left(\frac{x}{2}\right)+c$
B
$2 \tan \left(\frac{x}{2}\right)+c$
C
$-\cot \left(\frac{x}{2}\right)+c$
D
$-2 \cot \left(\frac{x}{2}\right)+c$

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Q 4MCQ2 Marks

If $\int \frac{d x}{4 x^2-1}=A \log \left(\frac{2 x-1}{2 x+1}\right)+c,$ then $A= ........ $

A
$1$
B
$\frac{1}{2}$
C
$\frac{1}{3}$
✓
$\frac{1}{4}$

Answer: D.

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Q 5MCQ2 Marks

$\int \frac{d x}{9 x^2+1}=\ldots \ldots .$

A
$\frac{1}{3} \tan ^{-1}(2 x)+ c$
B
$\frac{1}{3} \tan ^{-1} x+c$
C
$\frac{1}{3} \tan ^{-1}(3 x)+c$
D
$\frac{1}{3} \tan ^{-1}(6 x)+c$

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Q 6Answer the following questions in short.1 Mark

Evaluate : $\int \frac{\sin 2 x}{\cos x} d x$

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Q 7Solve the Following Question.(2 Marks)2 Marks

Evaluate: $\int \frac{1}{x \log x \log (\log x)} d x$

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Q 8Solve the Following Question.(2 Marks)2 Marks

Evaluate: $\int x \log x d x$

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Q 9Solve the Following Question.(2 Marks)2 Marks

Evaluate: $\int \frac{1}{x(x-1)} d x$

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Q 10Solve the Following Question.(2 Marks)2 Marks

Evaluate: $\int \frac{d x}{x^2+4 x+8}$

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Q 11Solve the Following Question.(2 Marks)2 Marks

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

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Q 12Solve the Following Question.(3 Marks)3 Marks

Evaluate : $\int \frac{1}{5-4 \cos x} d x$

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Q 13Solve the Following Question.(3 Marks)3 Marks

If $x=\phi(t)$ is a differentiable function of ' $t$ ', then prove that $\int f(x) d x=\int f[\phi(t)] \phi^{\prime}(t) d t$

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Q 14Solve the Following Question.(3 Marks)3 Marks

Evaluate : $\int \frac{d x}{2+\cos x-\sin x}$

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Q 15Solve the Following Question.(3 Marks)3 Marks

Evaluate: $\int \frac{x^2 \cdot \tan ^{-1}\left(x^3\right)}{1+x^6} d x$

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Q 16Solve the Following Question.(3 Marks)3 Marks

Evaluate: $\int \frac{1+\log x}{x(2+\log x)(3+\log x)} d x$

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Q 17Solve the Following Question.(4 Marks)4 Marks

If $u$ and $v$ are two differentiable functions of $x$ then prove that
$\int u \cdot v \cdot d x=u \int v \cdot d x-\int\left[\left(\frac{d u}{d x}\right) \int v \cdot d x\right] d x$
Hence evaluate $\int x \sin x d x$

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Q 18Solve the Following Question.(4 Marks)4 Marks

Show that: $\int \frac{1}{x^2 \sqrt{a^2+x}} d x=\frac{-1}{a^2} \frac{\sqrt{a^2+x^2}}{x}+c$

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Q 19Solve the Following Question.(4 Marks)4 Marks

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

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Q 20Solve the Following Question.(4 Marks)4 Marks

If $u$ and $v$ are two differentiable functions of $x$ then prove that :
$\int u v d x=u \int v d x-\int\left[\frac{d u}{d x} \int v d x\right] d x$
Hence evaluate $\int \log x d x$

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Q 21Solve the Following Question.(4 Marks)4 Marks

Prove that: $\int \sqrt{x^2+a^2} \cdot d x=\frac{x}{2} \sqrt{x^2+a^2}+\frac{a^2}{2} \log \left|x+\sqrt{x^2+a^2}\right|+c$ .

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