Calculus Examples

Evaluate the Integral integral of x^2e^(5x) with respect to x
Step 1
Integrate by parts using the formula , where and .
Step 2
Simplify.
Tap for more steps...
Step 2.1
Combine and .
Step 2.2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Combine and .
Step 5
Integrate by parts using the formula , where and .
Step 6
Simplify.
Tap for more steps...
Step 6.1
Combine and .
Step 6.2
Combine and .
Step 6.3
Combine and .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Let . Then , so . Rewrite using and .
Tap for more steps...
Step 8.1
Let . Find .
Tap for more steps...
Step 8.1.1
Differentiate .
Step 8.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 8.1.3
Differentiate using the Power Rule which states that is where .
Step 8.1.4
Multiply by .
Step 8.2
Rewrite the problem using and .
Step 9
Combine and .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Simplify.
Tap for more steps...
Step 11.1
Multiply by .
Step 11.2
Multiply by .
Step 12
The integral of with respect to is .
Step 13
Simplify.
Tap for more steps...
Step 13.1
Rewrite as .
Step 13.2
Simplify.
Tap for more steps...
Step 13.2.1
Combine and .
Step 13.2.2
Combine and .
Step 13.2.3
Combine and .
Step 13.2.4
Combine and .
Step 13.2.5
Combine and .
Step 13.2.6
To write as a fraction with a common denominator, multiply by .
Step 13.2.7
Combine and .
Step 13.2.8
Combine the numerators over the common denominator.
Step 13.2.9
Multiply by .
Step 13.2.10
Combine and .
Step 13.2.11
Multiply by .
Step 13.2.12
Cancel the common factor of and .
Tap for more steps...
Step 13.2.12.1
Factor out of .
Step 13.2.12.2
Cancel the common factors.
Tap for more steps...
Step 13.2.12.2.1
Factor out of .
Step 13.2.12.2.2
Cancel the common factor.
Step 13.2.12.2.3
Rewrite the expression.
Step 13.2.12.2.4
Divide by .
Step 14
Replace all occurrences of with .
Step 15
Reorder terms.