# Calculus Examples

Evaluate the Integral integral from 0 to 1 of cube root of 1+7x with respect to x
Step 1
Let . Then , so . Rewrite using and .
Step 1.1
Let . Find .
Step 1.1.1
Differentiate .
Step 1.1.2
Differentiate.
Step 1.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3
Evaluate .
Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Simplify.
Step 1.3.1
Multiply by .
Step 1.3.2
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Simplify.
Step 1.5.1
Multiply by .
Step 1.5.2
Step 1.6
The values found for and will be used to evaluate the definite integral.
Step 1.7
Rewrite the problem using , , and the new limits of integration.
Step 2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Use to rewrite as .
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Substitute and simplify.
Step 6.1
Evaluate at and at .
Step 6.2
Simplify.
Step 6.2.1
Rewrite as .
Step 6.2.2
Apply the power rule and multiply exponents, .
Step 6.2.3
Cancel the common factor of .
Step 6.2.3.1
Cancel the common factor.
Step 6.2.3.2
Rewrite the expression.
Step 6.2.4
Raise to the power of .
Step 6.2.5
Combine and .
Step 6.2.6
Multiply by .
Step 6.2.7
Cancel the common factor of and .
Step 6.2.7.1
Factor out of .
Step 6.2.7.2
Cancel the common factors.
Step 6.2.7.2.1
Factor out of .
Step 6.2.7.2.2
Cancel the common factor.
Step 6.2.7.2.3
Rewrite the expression.
Step 6.2.7.2.4
Divide by .
Step 6.2.8
One to any power is one.
Step 6.2.9
Multiply by .
Step 6.2.10
To write as a fraction with a common denominator, multiply by .
Step 6.2.11
Combine and .
Step 6.2.12
Combine the numerators over the common denominator.
Step 6.2.13
Simplify the numerator.
Step 6.2.13.1
Multiply by .
Step 6.2.13.2
Subtract from .
Step 6.2.14
Multiply by .
Step 6.2.15
Multiply by .
Step 7
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Step 8