Calculus Examples

Evaluate the Integral integral of 7xe^(5x^2) with respect to x
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Let . Then , so . Rewrite using and .
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Step 2.1
Let . Find .
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Step 2.1.1
Differentiate .
Step 2.1.2
Differentiate using the chain rule, which states that is where and .
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Step 2.1.2.1
To apply the Chain Rule, set as .
Step 2.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.1.2.3
Replace all occurrences of with .
Step 2.1.3
Differentiate.
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Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3.3
Multiply by .
Step 2.1.4
Simplify.
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Step 2.1.4.1
Reorder the factors of .
Step 2.1.4.2
Reorder factors in .
Step 2.2
Rewrite the problem using and .
Step 3
Apply the constant rule.
Step 4
Simplify the answer.
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Step 4.1
Rewrite as .
Step 4.2
Combine and .
Step 4.3
Replace all occurrences of with .