evaluate. (be sure to check by differentiating!) \n int(3 + x^{7})^{3}x^{6}mathrm{d}x \n int(3 +…

evaluate. (be sure to check by differentiating!) \n int(3 + x^{7})^{3}x^{6}mathrm{d}x \n int(3 + x^{7})^{3}x^{6}mathrm{d}x=square \n(type an exact answer. use parentheses to clearly denote the argument of each function.)

evaluate. (be sure to check by differentiating!) \n int(3 + x^{7})^{3}x^{6}mathrm{d}x \n int(3 + x^{7})^{3}x^{6}mathrm{d}x=square \n(type an exact answer. use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Use substitution

Let $u = 3 + x^{7}$, then $du=7x^{6}dx$, and $x^{6}dx=\frac{1}{7}du$.

Step2: Rewrite the integral

The integral $\int(3 + x^{7})^{3}x^{6}dx$ becomes $\int u^{3}\cdot\frac{1}{7}du=\frac{1}{7}\int u^{3}du$.

Step3: Integrate $u^{3}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\frac{1}{7}\cdot\frac{u^{4}}{4}+C=\frac{u^{4}}{28}+C$.

Step4: Substitute back $u$

Substitute $u = 3 + x^{7}$ back into the result, we get $\frac{(3 + x^{7})^{4}}{28}+C$.

Answer:

$\frac{(3 + x^{7})^{4}}{28}+C$