We can solve the integral $\int\sec\left(5w\right)\tan\left(5w\right)dw$ 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 $5w$ 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 $dw$ 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 $dw$ in the previous equation
Substituting $u$ and $dw$ in the integral and simplify
Take the constant $\frac{1}{5}$ out of the integral
Apply the formula: $\int\sec\left(\theta \right)\tan\left(\theta \right)dx$$=\sec\left(\theta \right)+C$, where $x=u$
Replace $u$ with the value that we assigned to it in the beginning: $5w$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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