Evaluate the Integral

Problem

(∫_^)(x2*sec2(x3)*d(x))

Solution

1. Identify the inner function for substitution, which is u=x3

2. Calculate the derivative of u to find d(u) resulting in d(u)=3*x2*d(x)

3. Rearrange the differential to solve for the terms present in the integral: 1/3*d(u)=x2*d(x)

4. Substitute the variables into the integral to rewrite it in terms of u

(∫_^)(sec2(u)⋅1/3*d(u))

5. Apply the constant multiple rule by moving the fraction outside the integral.

1/3*(∫_^)(sec2(u)*d(u))

6. Integrate using the standard trigonometric integral (∫_^)(sec2(u)*d(u))=tan(u)+C

1/3*tan(u)+C

7. Back-substitute the original expression x3 for u to get the final result.

Final Answer

(∫_^)(x2*sec2(x3)*d(x))=tan(x3)/3+C

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