Calculus Examples

Evaluate the Integral integral from 1 to 2 of 2xe^(-x^2) with respect to x
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Let . Then , so . Rewrite using and .
Tap for more steps...
Step 2.1
Let . Find .
Tap for more steps...
Step 2.1.1
Differentiate .
Step 2.1.2
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 2.1.2.1
To apply the Chain Rule, set as .
Step 2.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.1.2.3
Replace all occurrences of with .
Step 2.1.3
Differentiate.
Tap for more steps...
Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3.3
Multiply by .
Step 2.1.4
Simplify.
Tap for more steps...
Step 2.1.4.1
Reorder the factors of .
Step 2.1.4.2
Reorder factors in .
Step 2.2
Substitute the lower limit in for in .
Step 2.3
Simplify.
Tap for more steps...
Step 2.3.1
One to any power is one.
Step 2.3.2
Multiply by .
Step 2.3.3
Rewrite the expression using the negative exponent rule .
Step 2.4
Substitute the upper limit in for in .
Step 2.5
Simplify.
Tap for more steps...
Step 2.5.1
Raise to the power of .
Step 2.5.2
Multiply by .
Step 2.5.3
Rewrite the expression using the negative exponent rule .
Step 2.6
The values found for and will be used to evaluate the definite integral.
Step 2.7
Rewrite the problem using , , and the new limits of integration.
Step 3
Move the negative in front of the fraction.
Step 4
Apply the constant rule.
Step 5
Simplify the answer.
Tap for more steps...
Step 5.1
Combine and .
Step 5.2
Substitute and simplify.
Tap for more steps...
Step 5.2.1
Evaluate at and at .
Step 5.2.2
Simplify.
Tap for more steps...
Step 5.2.2.1
Rewrite as a product.
Step 5.2.2.2
Multiply by .
Step 5.2.2.3
Move to the left of .
Step 5.2.2.4
Rewrite as a product.
Step 5.2.2.5
Multiply by .
Step 5.2.2.6
Move to the left of .
Step 6
Simplify.
Tap for more steps...
Step 6.1
Apply the distributive property.
Step 6.2
Cancel the common factor of .
Tap for more steps...
Step 6.2.1
Move the leading negative in into the numerator.
Step 6.2.2
Factor out of .
Step 6.2.3
Cancel the common factor.
Step 6.2.4
Rewrite the expression.
Step 6.3
Cancel the common factor of .
Tap for more steps...
Step 6.3.1
Factor out of .
Step 6.3.2
Cancel the common factor.
Step 6.3.3
Rewrite the expression.
Step 6.4
Move the negative in front of the fraction.
Step 7
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 8