Learn how to solve differential equations problems step by step online. Integrate the function (1+e^tan(2x))sec(2x)^2 from 0 to pi/8. We can solve the integral \int_{0}^{\frac{\pi }{8}}\left(1+e^{\tan\left(2x\right)}\right)\sec\left(2x\right)^2dx by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that \tan\left(2x\right) it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part. Now, in order to rewrite dx in terms of du, we need to find the derivative of u. We need to calculate du, we can do that by finding the derivative of the equation above. Isolate dx in the previous equation. Substituting u and dx in the integral and simplify.

Integrate the function (1+e^tan(2x))sec(2x)^2 from 0 to pi/8