Calculus Examples

Evaluate the Integral integral of xarctan(x^2) with respect to x
Step 1
Let . Then , so . Rewrite using and .
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Step 1.1
Let . Find .
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Step 1.1.1
Differentiate .
Step 1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2
Rewrite the problem using and .
Step 2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Integrate by parts using the formula , where and .
Step 5
Combine and .
Step 6
Let . Then , so . Rewrite using and .
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Step 6.1
Let . Find .
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Step 6.1.1
Differentiate .
Step 6.1.2
By the Sum Rule, the derivative of with respect to is .
Step 6.1.3
Differentiate using the Power Rule which states that is where .
Step 6.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.5
Add and .
Step 6.2
Rewrite the problem using and .
Step 7
Simplify.
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Step 7.1
Multiply by .
Step 7.2
Move to the left of .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
The integral of with respect to is .
Step 10
Simplify.
Step 11
Substitute back in for each integration substitution variable.
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Step 11.1
Replace all occurrences of with .
Step 11.2
Replace all occurrences of with .
Step 11.3
Replace all occurrences of with .
Step 12
Simplify.
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Step 12.1
Simplify each term.
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Step 12.1.1
Multiply the exponents in .
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Step 12.1.1.1
Apply the power rule and multiply exponents, .
Step 12.1.1.2
Multiply by .
Step 12.1.2
Combine and .
Step 12.2
To write as a fraction with a common denominator, multiply by .
Step 12.3
Combine and .
Step 12.4
Combine the numerators over the common denominator.
Step 12.5
Move to the left of .
Step 12.6
Multiply .
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Step 12.6.1
Multiply by .
Step 12.6.2
Multiply by .
Step 12.7
Reorder factors in .
Step 13
Reorder terms.