Calculus Examples

Evaluate the Integral integral of (x^3)/( square root of 1+x^2) with respect to x
Step 1
Let , where . Then . Note that since , is positive.
Step 2
Simplify terms.
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Step 2.1
Simplify .
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Step 2.1.1
Rearrange terms.
Step 2.1.2
Apply pythagorean identity.
Step 2.1.3
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2
Cancel the common factor of .
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Step 2.2.1
Factor out of .
Step 2.2.2
Cancel the common factor.
Step 2.2.3
Rewrite the expression.
Step 3
Raise to the power of .
Step 4
Factor out .
Step 5
Using the Pythagorean Identity, rewrite as .
Step 6
Simplify.
Step 7
Let . Then , so . Rewrite using and .
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Step 7.1
Let . Find .
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Step 7.1.1
Differentiate .
Step 7.1.2
The derivative of with respect to is .
Step 7.2
Rewrite the problem using and .
Step 8
Split the single integral into multiple integrals.
Step 9
Apply the constant rule.
Step 10
By the Power Rule, the integral of with respect to is .
Step 11
Simplify.
Step 12
Substitute back in for each integration substitution variable.
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Step 12.1
Replace all occurrences of with .
Step 12.2
Replace all occurrences of with .