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.

Step-by-step explanation:

To integrate the term, substitute,

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

On differentiating equation (1),

`\begin{gathered} (du)/(dx)=4(1) \\ dx=(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 ((u-7)/(4))u^8((1)/(4))du \\ =(1)/(16)\int (u-7)u^8du \\ =(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=(1)/(16)\lbrack(u^(10))/(10)-(7u^9)/(9)\rbrack \\ =(u^(10))/(160)-(7u^9)/(144) \end{gathered}`

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

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

Hence, the value of the integral is obtained.