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Chapter 5: Problem 94

In Exercises \(91-108,\) find the indefinite integral. $$\int(2 x+1) e^{x^{2}+x} d x$$

Short Answer

Expert verified
\[e^{x^{2}+x}+C\]

Step by step solution

01

Identify the substitution

Set \(u=x^{2}+x\). This part of the function will be replaced in the next steps.
02

Find the derivative of u

Now, calculate the derivative of \(u\), which is \(du=(2x+1) dx\). This form will allow to replace \(dx\) in the integral.
03

Perform the substitution

Replace \(x^{2}+x\) with \(u\) and \(dx\) with \(du/(2x+1)\) to rewrite the integral as \(\int e^u du\).
04

Evaluate the integral

The integral of \(e^u\) is \(e^u\). So, the solution to the integral is \(e^u+C\), where \(C\) is the constant of integration.
05

Substitute back

Replace \(u\) with \(x^{2}+x\) to get the result as \(e^{x^{2}+x}+C\). This is the final answer.

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