Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating

$\int {\left(x^3+4\right)}^{2}dx$.

Short answer

The answer is$\frac{x^7}{7}+2x^4+16x+C$.

Step-by-step solution

Step 1. Given Information.

The given integral is$\int {\left(x^3+4\right)}^{2}dx.$

Step 2. Integration.

On Integrating the function,

$$\begin{gathered} \int {\left(x^3+4\right)}^{2}dx \\ =\int \left(x^6+8x^3+16\right)dx \\ =\int x^{6}dx+\int 8x^{3}dx+\int 16dx \\ =\left(\frac{x^7}{7}\right)+8\left(\frac{x^4}{4}\right)+16x+C \\ =\frac{x^7}{7}+2x^4+16x+C \end{gathered}$$

Step 3. Verification.

On Differentiating $\frac{x^7}{7}+2x^4+16x$, we get,

localid="1648577234721" $$\begin{gathered} =\frac{7x^6}{7}+8x^3+17 \\ =x^6+8x^3+17 \\ =x^6+8x^3+16+1 \\ ={\left(x^3+4\right)}^2+C \end{gathered}$$

Hence Proved.