Question

Evaluate the indefinite integral. (Use C for the constant of integration.) x(4x + 7)8dxEvaluate the indefinite integral. (Use C for the constant of integration.) x(4x + 7)^8dx

Answer 1

Given:

The indefinite integral is given as,

$\int x(4x+7)^8_{}dx$

The objective is to evaluate the integral.

Explanation:

To integrate the term, substitute,

$\begin{gathered} u=4x+7\text{ . . . . . . . (1)} \\ x=\frac{u-7}{4}\text{ . .. . }\ldots\text{ ..(2)} \end{gathered}$

On differentiating equation (1),

$\begin{gathered} \frac{du}{dx}=4(1) \\ dx=\frac{1}{4}du\text{ . . . . . . . .(3)} \end{gathered}$

Substitute equations (1), (2), and (3) in the given expression.

$\begin{gathered} \int x(4x+7)^8_{}dx=\int (\frac{u-7}{4})u^8(\frac{1}{4})du \\ =\frac{1}{16}\int (u-7)u^8du \\ =\frac{1}{16}\int (u^9-7u^8)du \end{gathered}$

On further integrating the above expression using the power rule,

$\begin{gathered} \int x(4x+7)^8_{}dx=\frac{1}{16}\lbrack\frac{u^{10}}{10}-\frac{7u^9}{9}\rbrack \\ =\frac{u^{10}}{160}-\frac{7u^9}{144} \end{gathered}$

Now, replace the value of u in the above expression.

$\int x(4x+7)^8_{}dx=\frac{(4x+7)^{10}}{160}-\frac{7(4x+7)^9^{}}{144}+C$

Hence, the value of the integral is obtained.