Chapter 5.5, Problem 23E

To determine

To evaluate: The indefinite integral.

Answer to Problem 23E

The integration of the indefinite integral ${\displaystyle \int \left(x^2+1\right){\left(x^3+3x\right)}^4}dx$is $\frac{1}{15}{\left(x^3+3x\right)}^5+C$ .

Explanation of Solution

Given information: ${\displaystyle \int \left(x^2+1\right){\left(x^3+3x\right)}^4}dx$

Calculation: ${\displaystyle \int \left(x^2+1\right){\left(x^3+3x\right)}^4}dx$

Let apply the substitution $z=x^3+3x$

Partially differentiate both side,

  $\begin{array}{l}\Rightarrow dz=3x^{2}dx+3dx\\ \Rightarrow dz=3\left(x^2+1\right)dx\\ \Rightarrow \frac{dz}{3}=\left(x^2+1\right)dx\end{array}$

Let’s evaluate the indefinite integral,

  $\begin{array}{l}={\displaystyle \int \frac{z^4}{3}dz}\\ =\frac{1}{3}{\displaystyle \int z^{4}dz}\end{array}$

Therefore, using the power rule,

  $\begin{array}{l}=\frac{1}{3}\frac{z^{4+1}}{4+1}\\ =\frac{1}{3}\frac{z^5}{5}\\ =\frac{1}{15}{z}^5\end{array}$

Now, Substitute back $z=x^3+3x$ ,

  $=\frac{1}{15}{\left(x^3+3x\right)}^5$

Adding a constant with the solution,

  $=\frac{1}{15}{\left(x^3+3x\right)}^5+C$

Hence, The integration of the indefinite integral ${\displaystyle \int \left(x^2+1\right){\left(x^3+3x\right)}^4}dx$is $\frac{1}{15}{\left(x^3+3x\right)}^5+C$ .