The value of integral {d x}{ ² x+ ² x} is...

Correct Answer is : −12tan⁡x−122tan⁡−1(tan⁡x2)+c-1/2 x-{1}{2 {2}} ^{-1}≤ft({ x}{{2}})+c2tanx−1−221tan−1(2tanx)+c

-1/2 x+{1}{2 {2}} ^{-1}≤ft({ x}{{2}})+c

-1/2 x-{1}{2 {2}} ^{-1}≤ft({ x}{{2}})+c

1/2 x-{1}{2 {2}} ^{-1}≤ft({ x}{{2}})+c

1/2 x+{1}{2 {2}} ^{-1}≤ft({ x}{{2}})+c

Correct Answer is : −12tan⁡x−122tan⁡−1(tan⁡x2)+c-1/2 x-{1}{2 {2}} ^{-1}≤ft({ x}{{2}})+c2tanx−1−221tan−1(2tanx)+c

Test Index

MHTCET 19 Apr 2026 Shift 1 Paper

Question : 128 of 150

Marks: +1, -0

The value of integral

\int \frac{d x}{\sin ^{2} x+\tan ^{2} x}

is...

$\frac{-1}{2 \tan x}+\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+c$
$\frac{-1}{2 \tan x}-\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+c$
$\frac{1}{2 \tan x}-\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+c$
$\frac{1}{2 \tan x}+\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+c$

Solution:

Concept:

Convert the integral to a function of

\tan x

using the substitution

t = \tan x

, then apply partial fractions and standard integration formulas.

Explanation:

Let

t = \tan x

Then

dx = \frac{dt}{1+t^2}

and

\sin^2 x = \frac{t^2}{1+t^2}

So the denominator becomes:

\sin^2 x + \tan^2 x = \frac{t^2}{1+t^2} + t^2 = \frac{t^2(1+t^2) + t^2}{1+t^2} = \frac{t^2(t^2+2)}{1+t^2}.

Thus the integral simplifies to:

\int \frac{dx}{\sin^2 x + \tan^2 x} = \int \frac{dt}{t^2(t^2+2)}.

Using partial fractions:

\frac{1}{t^2(t^2+2)} = \frac{1}{2}\left(\frac{1}{t^2} - \frac{1}{t^2+2}\right).

Integrating term by term:

\frac{1}{2}\int \frac{dt}{t^2} - \frac{1}{2}\int \frac{dt}{t^2+2} = -\frac{1}{2t} - \frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{t}{\sqrt{2}}\right) + C.

Substitute back

t = \tan x

\int \frac{dx}{\sin^2 x + \tan^2 x} = -\frac{1}{2\tan x} - \frac{1}{2\sqrt{2}} \tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right) + C.

Answer:

Option B:

\frac{-1}{2 \tan x} - \frac{1}{2 \sqrt{2}} \tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right) + c

Go to Question:

Prev Question

Next Question