hw part 1: 7.2 trigonometric integrals\nscore: 1.4/8 answered: 2/8\nprogress saved done\nquestion 3\n0/1 pt…

Explanation:

Step1: Use double - angle formula

We know that $\cos^{2}\theta=\frac{1 + \cos(2\theta)}{2}$. Here $\theta = 5x$, so $\cos^{2}(5x)=\frac{1+\cos(10x)}{2}$. Then the integral becomes $\int\frac{1+\cos(10x)}{2}dx$.

Step2: Split the integral

$\int\frac{1+\cos(10x)}{2}dx=\frac{1}{2}\int(1 + \cos(10x))dx=\frac{1}{2}\left(\int 1dx+\int\cos(10x)dx\right)$.

Step3: Integrate each term

$\int 1dx=x$ and for $\int\cos(10x)dx$, let $u = 10x$, then $du=10dx$ and $\int\cos(10x)dx=\frac{1}{10}\int\cos(u)du=\frac{1}{10}\sin(u)=\frac{1}{10}\sin(10x)$.

Step4: Combine the results

$\frac{1}{2}\left(\int 1dx+\int\cos(10x)dx\right)=\frac{1}{2}\left(x+\frac{1}{10}\sin(10x)\right)=\frac{x}{2}+\frac{\sin(10x)}{20}$.

Answer:

$\frac{x}{2}+\frac{\sin(10x)}{20}$