"Solve this equation please " was answered by our community

High School
Calculus and Analysis
Calculus 1
Integrals

daet.nathaniel74

Answered question

2022-02-17

Solve this equation please

Answer & Explanation

nick1337

Expert2023-04-23Added 777 answers

To solve this integral, we can use substitution. Let's substitute $u = \tan (5 w)$ , then $\frac{d u}{d w} = 5 \sec^{2} (5 w) d w, and d w = \frac{d u}{5 \sec^{2} (5 w)}$ .

Substituting these values into the integral, we get:
$\int (\sec^{2} (5 w) e^{2 (\tan (5 w))}) d w$
$= \int (e^{2 u} \frac{d u}{5})$
$= (\frac{1}{5}) \int (e^{2 u}) d u$
$= (\frac{1}{5}) (\frac{1}{2}) e^{2 u} + C$
$= (\frac{1}{10}) e^{2 \tan (5 w)} + C$

So the final answer is:
$(\frac{1}{10}) e^{2 \tan (5 w)} + C$

Note that C is the constant of integration, which can take any value.

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