Calculus Examples

Evaluate the Integral integral of x square root of x-1 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
Let . Then . Rewrite using and .
Tap for more steps...
Step 4.1
Let . Find .
Tap for more steps...
Step 4.1.1
Differentiate .
Step 4.1.2
By the Sum Rule, the derivative of with respect to is .
Step 4.1.3
Differentiate using the Power Rule which states that is where .
Step 4.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.5
Add and .
Step 4.2
Rewrite the problem using and .
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Simplify.
Tap for more steps...
Step 6.1
Rewrite as .
Step 6.2
Simplify.
Tap for more steps...
Step 6.2.1
Combine and .
Step 6.2.2
Combine and .
Step 6.2.3
Multiply by .
Step 6.2.4
Multiply by .
Step 6.2.5
Multiply by .
Step 6.2.6
To write as a fraction with a common denominator, multiply by .
Step 6.2.7
Combine and .
Step 6.2.8
Combine the numerators over the common denominator.
Step 6.2.9
Combine and .
Step 6.2.10
Multiply by .
Step 6.2.11
Combine and .
Step 6.2.12
Multiply by .
Step 6.2.13
Factor out of .
Step 6.2.14
Cancel the common factors.
Tap for more steps...
Step 6.2.14.1
Factor out of .
Step 6.2.14.2
Cancel the common factor.
Step 6.2.14.3
Rewrite the expression.
Step 6.2.15
Move the negative in front of the fraction.
Step 6.2.16
To write as a fraction with a common denominator, multiply by .
Step 6.2.17
Combine and .
Step 6.2.18
Combine the numerators over the common denominator.
Step 6.2.19
Multiply by .
Step 6.2.20
Rewrite as a product.
Step 6.2.21
Multiply by .
Step 6.2.22
Multiply by .
Step 7
Replace all occurrences of with .
Step 8
Reorder terms.