Calculus Examples

Evaluate the Integral integral of (x^3)/( square root of x^2+36) with respect to x
Step 1
Let , where . Then . Note that since , is positive.
Step 2
Simplify terms.
Tap for more steps...
Step 2.1
Simplify .
Tap for more steps...
Step 2.1.1
Simplify each term.
Tap for more steps...
Step 2.1.1.1
Apply the product rule to .
Step 2.1.1.2
Raise to the power of .
Step 2.1.2
Factor out of .
Tap for more steps...
Step 2.1.2.1
Factor out of .
Step 2.1.2.2
Factor out of .
Step 2.1.2.3
Factor out of .
Step 2.1.3
Apply pythagorean identity.
Step 2.1.4
Rewrite as .
Step 2.1.5
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2
Reduce the expression by cancelling the common factors.
Tap for more steps...
Step 2.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.2.1.1
Factor out of .
Step 2.2.1.2
Cancel the common factor.
Step 2.2.1.3
Rewrite the expression.
Step 2.2.2
Simplify.
Tap for more steps...
Step 2.2.2.1
Factor out of .
Step 2.2.2.2
Apply the product rule to .
Step 2.2.2.3
Raise to the power of .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Raise to the power of .
Step 5
Factor out .
Step 6
Using the Pythagorean Identity, rewrite as .
Step 7
Simplify.
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
The derivative of with respect to is .
Step 8.2
Rewrite the problem using and .
Step 9
Split the single integral into multiple integrals.
Step 10
Apply the constant rule.
Step 11
By the Power Rule, the integral of with respect to is .
Step 12
Simplify.
Tap for more steps...
Step 12.1
Combine and .
Step 12.2
Simplify.
Step 13
Substitute back in for each integration substitution variable.
Tap for more steps...
Step 13.1
Replace all occurrences of with .
Step 13.2
Replace all occurrences of with .
Step 14
Simplify.
Tap for more steps...
Step 14.1
Simplify each term.
Tap for more steps...
Step 14.1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 14.1.2
Apply the product rule to .
Step 14.1.3
Raise to the power of .
Step 14.1.4
Write as a fraction with a common denominator.
Step 14.1.5
Combine the numerators over the common denominator.
Step 14.1.6
Rewrite as .
Tap for more steps...
Step 14.1.6.1
Factor the perfect power out of .
Step 14.1.6.2
Factor the perfect power out of .
Step 14.1.6.3
Rearrange the fraction .
Step 14.1.7
Pull terms out from under the radical.
Step 14.1.8
Combine and .
Step 14.1.9
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 14.1.10
Apply the product rule to .
Step 14.1.11
Raise to the power of .
Step 14.1.12
Write as a fraction with a common denominator.
Step 14.1.13
Combine the numerators over the common denominator.
Step 14.1.14
Rewrite as .
Tap for more steps...
Step 14.1.14.1
Factor the perfect power out of .
Step 14.1.14.2
Factor the perfect power out of .
Step 14.1.14.3
Rearrange the fraction .
Step 14.1.15
Pull terms out from under the radical.
Step 14.1.16
Combine and .
Step 14.1.17
Apply the product rule to .
Step 14.1.18
Combine.
Step 14.1.19
Multiply by .
Step 14.1.20
Raise to the power of .
Step 14.1.21
Simplify the numerator.
Tap for more steps...
Step 14.1.21.1
Rewrite as .
Step 14.1.21.2
Factor out .
Step 14.1.21.3
Pull terms out from under the radical.
Step 14.1.21.4
Apply the distributive property.
Step 14.1.21.5
Factor out of .
Tap for more steps...
Step 14.1.21.5.1
Factor out of .
Step 14.1.21.5.2
Factor out of .
Step 14.1.21.5.3
Factor out of .
Step 14.1.22
Multiply by .
Step 14.2
To write as a fraction with a common denominator, multiply by .
Step 14.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps...
Step 14.3.1
Multiply by .
Step 14.3.2
Multiply by .
Step 14.4
Combine the numerators over the common denominator.
Step 14.5
Cancel the common factor of .
Tap for more steps...
Step 14.5.1
Factor out of .
Step 14.5.2
Cancel the common factor.
Step 14.5.3
Rewrite the expression.
Step 14.6
Simplify the numerator.
Tap for more steps...
Step 14.6.1
Factor out of .
Tap for more steps...
Step 14.6.1.1
Factor out of .
Step 14.6.1.2
Factor out of .
Step 14.6.2
Multiply by .
Step 14.6.3
Add and .
Step 14.7
Rewrite as .
Step 14.8
Factor out of .
Step 14.9
Factor out of .
Step 14.10
Move the negative in front of the fraction.