Question

$int (x^{5}+x^{3})^{11}(5x^{4}+3x^{2})mathrm{d}x=square$

Explanation:

Step1: Set substitution

Let $u = x^{5}+x^{3}$, then $du=(5x^{4}+3x^{2})dx$.

Step2: Rewrite the integral

The original integral $\int(x^{5}+x^{3})^{11}(5x^{4}+3x^{2})dx$ becomes $\int u^{11}du$.

Step3: Integrate $u^{11}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), we have $\int u^{11}du=\frac{u^{12}}{12}+C$.

Step4: Substitute back $u$

Substitute $u = x^{5}+x^{3}$ back into the result, we get $\frac{(x^{5}+x^{3})^{12}}{12}+C$.

Answer:

$\frac{(x^{5}+x^{3})^{12}}{12}+C$