sec^2(x) / (1+2tanx)^2 dx

please help me with this integration thank you

hint:

What would you get if you differentiated

(1 + 2tanx)^-1

where did the (1 + 2tanx)^-1

come from?

To integrate the given expression, sec^2(x) / (1+2tan(x))^2 dx, we can use the substitution method.

Let's start by substituting u = tan(x). Then, we can rewrite sec^2(x) using the trigonometric identity: sec^2(x) = 1 + tan^2(x).

Now, differentiate both sides of the substitution equation with respect to x to find du/dx:

du/dx = d(tan(x))/dx = sec^2(x)

Rearranging the equation, we have dx = du / sec^2(x).

Substituting these values back into the integral, we get:

∫ sec^2(x) / (1 + 2tan(x))^2 dx
= ∫ (1 + tan^2(x)) / (1 + 2tan(x))^2 dx
= ∫ (1 + u^2) / (1 + 2u)^2 * (du / sec^2(x))

Now, we can simplify further by canceling out sec^2(x) in the integral:

= ∫ (1 + u^2) * cos^2(x) / (1 + 2u)^2 du

The next step is to determine the new limits of integration. Since we made the substitution u = tan(x), we need to rewrite the limits of integration as well.

When x = 0, u = tan(0) = 0. So, the lower limit becomes u = 0.

When x = π/4, u = tan(π/4) = 1. So, the upper limit becomes u = 1.

Now, we can proceed to integrate the simplified expression:

∫ (1 + u^2) * cos^2(x) / (1 + 2u)^2 du
= ∫ (1 + u^2) * cos^2(x) / (1 + 2u)^2 du

To integrate this expression, we can use the method of partial fractions or trigonometric identities, depending on the form of the integral.

Once the integration is performed, we substitute back u = tan(x) and simplify the final result.