Q. 51 - Calculus

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. (Hint for Exercise 54: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$\int x^2 {\left(x^3+1\right)}^{5}dx.$

Step-by-Step Solution

Answer

The value of given integral is $\frac{{\left(x^3+1\right)}^6}{18} + c.$

Step 1. Given Information.

Given an integral$\int x^2 {\left(x^3+1\right)}^{5}dx.$

Step 2. Formula involved.

We use substitution method in this integration and then below formula will be used:

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

Step 3. Solving the integral.

$$\begin{gathered} Let t = x^3+1, \\ dt = 3x^{2}dx, or x^{2}dx = \frac{dt}{3}. \\ Subsituting this in the integral we get, \\ \int x^2 {\left(x^3+1\right)}^{5}dx \\ =\int t^5\frac{dt}{3} = \frac{1}{3}\int t^{5}dt \\ =\frac{1}{3}\frac{t^6}{6} +c = \frac{t^6}{18}+c = \frac{{\left(x^3+1\right)}^6}{18}+c. \end{gathered}$$