Calculus Examples

Evaluate the Integral integral from 0 to 6 of x/( square root of 9+2x^2) with respect to x
Step 1
Let . Then , so . Rewrite using and .
Tap for more steps...
Step 1.1
Let . Find .
Tap for more steps...
Step 1.1.1
Differentiate .
Step 1.1.2
Differentiate.
Tap for more steps...
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 .
Tap for more steps...
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
Add and .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Simplify.
Tap for more steps...
Step 1.3.1
Simplify each term.
Tap for more steps...
Step 1.3.1.1
Raising to any positive power yields .
Step 1.3.1.2
Multiply by .
Step 1.3.2
Add and .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Simplify.
Tap for more steps...
Step 1.5.1
Simplify each term.
Tap for more steps...
Step 1.5.1.1
Raise to the power of .
Step 1.5.1.2
Multiply by .
Step 1.5.2
Add and .
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
Simplify.
Tap for more steps...
Step 2.1
Multiply by .
Step 2.2
Move to the left of .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Apply basic rules of exponents.
Tap for more steps...
Step 4.1
Use to rewrite as .
Step 4.2
Move out of the denominator by raising it to the power.
Step 4.3
Multiply the exponents in .
Tap for more steps...
Step 4.3.1
Apply the power rule and multiply exponents, .
Step 4.3.2
Combine and .
Step 4.3.3
Move the negative in front of the fraction.
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Substitute and simplify.
Tap for more steps...
Step 6.1
Evaluate at and at .
Step 6.2
Simplify.
Tap for more steps...
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 .
Tap for more steps...
Step 6.2.3.1
Cancel the common factor.
Step 6.2.3.2
Rewrite the expression.
Step 6.2.4
Evaluate the exponent.
Step 6.2.5
Multiply by .
Step 6.2.6
Rewrite as .
Step 6.2.7
Apply the power rule and multiply exponents, .
Step 6.2.8
Cancel the common factor of .
Tap for more steps...
Step 6.2.8.1
Cancel the common factor.
Step 6.2.8.2
Rewrite the expression.
Step 6.2.9
Evaluate the exponent.
Step 6.2.10
Multiply by .
Step 6.2.11
Subtract from .
Step 6.2.12
Combine and .
Step 6.2.13
Cancel the common factor of and .
Tap for more steps...
Step 6.2.13.1
Factor out of .
Step 6.2.13.2
Cancel the common factors.
Tap for more steps...
Step 6.2.13.2.1
Factor out of .
Step 6.2.13.2.2
Cancel the common factor.
Step 6.2.13.2.3
Rewrite the expression.
Step 6.2.13.2.4
Divide by .
Step 7