Question

evaluate the integral.
int 7xcot x^{2}dx
int 7xcot x^{2}dx=square
(use parentheses to clearly denote the argument of each function)

Explanation:

Step1: Use substitution

Let $u = x^{2}$, then $du=2x\ dx$, and $x\ dx=\frac{1}{2}du$. The integral $\int7x\cot(x^{2})dx$ becomes $\frac{7}{2}\int\cot(u)du$.

Step2: Recall the integral of cotangent

We know that $\int\cot(u)du=\ln|\sin(u)| + C$. So, $\frac{7}{2}\int\cot(u)du=\frac{7}{2}\ln|\sin(u)|+C$.

Step3: Substitute back $u = x^{2}$

Substituting $u = x^{2}$ back into the result, we get $\frac{7}{2}\ln|\sin(x^{2})|+C$.

Answer:

$\frac{7}{2}\ln|\sin(x^{2})|+C$