Find out the answer to "Solve the integral by substitution. ∫sec2(arctan⁡x)1+x2dx"

Guenuegoomyns

Answered question

2021-11-09

Solve the integral by substitution.

\int \frac{\sec^{2} (\arctan x)}{1 + x^{2}} d x

Oung1985

Beginner2021-11-10Added 16 answers

Step 1
We have to use integration by substitution technique. we can use substitution for

\arctan x

.
Step 2
Consider the integral

\int \frac{\sec^{2} (\arctan x)}{1 + x^{2}} d x

. We can use substitution for

\arctan x

.
let

\arctan x = u

d u = \frac{1}{1 + x^{2}} d x

substituting the above values in the integral, we get

\int \frac{\sec^{2} (\arctan x)}{1 + x^{2}} d x = \int \sec^{2} u d u

= \tan u + c

we know that

\int \sec^{2} x = \tan x + c

substitute

u = \arctan x

in the above integrated value, we get

\int \frac{\sec^{2} (\arctan x)}{1 + x^{2}} d x = \tan (\arctan x) + c

=x+c

\tan and \arctan

are inverses.
hence we have

\int \frac{\sec^{2} (\arctan x)}{1 + x^{2}} d x = x + c

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