Step by step solution to "Integrate tan x sec^2 x dx?"

Mara Boyd

Answered question

2023-01-20

Integrate tan x sec^2 x dx?

wolfnightswimfdi

Beginner2023-01-21Added 9 answers

Given:

\int \tan x \sec^{2} (x) d x

Let's use the u-substitution given below:

u = \tan x

Differentiate

u :

\frac{d u}{d x} = \sec^{2} (x) d x

Let's multiply both sides of this derivative by to create a differential

d x :

d u = \sec^{2} (x) d x

We see du appears in our integral; therefore, this substitution works. We replace

\tan x

with u and

\sec^{2} (x) d x

with du:

\int \tan x \sec^{2} (x) d x = \int u d u

Recall that

\int x^{a} d x = \frac{x^{a + 1}}{a + 1} + C

Where C is the constant of integration.
Then,

\int u d u = \int u^{1} d u = \frac{u^{1 + 1}}{1 + 1} + C = \frac{u^{2}}{2} + C

Rewrite in terms of x. Since

u = \tan x, u^{2} = \tan^{2} (x) :

\frac{u^{2}}{2} + C = \frac{\tan^{2} (x)}{2} + C

Thus,

\int \tan x \sec^{2} (x) d x = \frac{\tan^{2} (x)}{2} + C

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