Chapter 5.2, Problem 1E

To determine

To Calculate: ${\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}$ using substitution $u=\left(2x^2+3\right)$

Answer to Problem 1E

  ${\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}=\frac{1}{44}{\left(2x^2+3\right)}^{11}+C$

Explanation of Solution

Given information:

  ${\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}$ , $u=\left(2x^2+3\right)$

Concept and Formula Used:

  ${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$ .

Calculation:

It is given that

  ${\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}$

Let $u=\left(2x^2+3\right)$

Differentiate both sides with respect to ‘x’

  $\begin{array}{l}\frac{du}{dx}=\frac{d}{dx}\left(2x^2+3\right)\\ \frac{du}{dx}=4x\\ du=4xdx\\ xdx=\frac{1}{4}du\end{array}$

Putting the value

  $\begin{array}{l}{\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}=\frac{1}{4}{\displaystyle \int {\left(u\right)}^{10}du}\\ \text{Using }{\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}\\ {\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}=\frac{1}{4}\left(\frac{u^{10+1}}{10+1}\right)+C\\ {\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}=\frac{1}{44}{u}^{11}+C\\ \text{Putting }u=\left(2x^2+3\right)\\ {\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}=\frac{1}{44}{\left(2x^2+3\right)}^{11}+C\end{array}$

Conclusion:

  ${\displaystyle \int x{\left(2x^2+3\right)}^{10}dx}=\frac{1}{44}{\left(2x^2+3\right)}^{11}+C$