Question

Integrate tan x sec^2 x .dx?

Answer 1

`inttanxsec^2(x)dx=tan^2(x)/2+C`

Explanation:

السلام عليكم.

We have

`inttanxsec^2(x)dx`

Let's make the following `u`-substitution:

`u=tanx`

Differentiate `u:`

`(du)/dx=sec^2(x)dx`

Let's turn this derivative into a differential by multiplying both sides by `dx:`

`du=sec^2(x)dx`

We see `du` appears in our integral; therefore, this substitution works. We replace `tanx` with `u` and `sec^2(x)dx` with `du:`

`inttanxsec^2(x)dx=intudu`

Recall that `intx^adx=x^(a+1)/(a+1)+C` Where `C` is the constant of integration.

Therefore,

`intudu=intu^1du=u^(1+1)/(1+1)+C=u^2/2+C`

Rewrite in terms of `x.` Since `u=tanx, u^2=tan^2(x):`

`u^2/2+C=tan^2(x)/2+C`

Thus,

`inttanxsec^2(x)dx=tan^2(x)/2+C`