We can solve the integral $\int\sec\left(2x\right)^5dx$ 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 $2x$ 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
$du=2dx$
Intermediate steps
3
Isolate $dx$ in the previous equation
$dx=\frac{du}{2}$
4
Substituting $u$ and $dx$ in the integral and simplify
$\int\frac{\sec\left(u\right)^5}{2}du$
5
Take the constant $\frac{1}{2}$ out of the integral
Solve the product $\frac{1}{2}\left(\frac{1}{4}\sec\left(u\right)^3\tan\left(u\right)+\frac{3}{8}\sec\left(u\right)\tan\left(u\right)+\frac{3}{8}\ln\left|\sec\left(u\right)+\tan\left(u\right)\right|\right)$
Solve the product $\frac{1}{2}\left(\frac{3}{8}\sec\left(u\right)\tan\left(u\right)+\frac{3}{8}\ln\left|\sec\left(u\right)+\tan\left(u\right)\right|\right)$
