Calculus Examples

Evaluate the Integral integral from 0.3 to 1.1 of x^3cot(x^4) 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
Substitute the lower limit in for in .
Step 1.3
Raise to the power of .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Raise to the power of .
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.
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Step 2.1
Rewrite as .
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Step 2.1.1
Use to rewrite as .
Step 2.1.2
Apply the power rule and multiply exponents, .
Step 2.1.3
Combine and .
Step 2.1.4
Cancel the common factor of and .
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Step 2.1.4.1
Factor out of .
Step 2.1.4.2
Cancel the common factors.
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Step 2.1.4.2.1
Factor out of .
Step 2.1.4.2.2
Cancel the common factor.
Step 2.1.4.2.3
Rewrite the expression.
Step 2.1.4.2.4
Divide by .
Step 2.2
Combine and .
Step 2.3
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Let . Then , so . Rewrite using and .
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Step 4.1
Let . Find .
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Step 4.1.1
Differentiate .
Step 4.1.2
Differentiate using the Power Rule which states that is where .
Step 4.2
Substitute the lower limit in for in .
Step 4.3
Raise to the power of .
Step 4.4
Substitute the upper limit in for in .
Step 4.5
Raise to the power of .
Step 4.6
The values found for and will be used to evaluate the definite integral.
Step 4.7
Rewrite the problem using , , and the new limits of integration.
Step 5
Combine and .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Simplify.
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Step 7.1
Multiply by .
Step 7.2
Multiply by .
Step 8
The integral of with respect to is .
Step 9
Evaluate at and at .
Step 10
Simplify.
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Step 10.1
Use the quotient property of logarithms, .
Step 10.2
Combine and .
Step 11
The result can be shown in multiple forms.
Exact Form:
Decimal Form: