Evaluate the Integral integral of x^3e^(x^2) with respect to x

Problem

(∫_^)(x3*e(x2)*d(x))

Solution

1. Substitute a new variable to simplify the exponent by letting u=x2

2. Differentiate the substitution to find d(u)=2*x*d(x) which implies x*d(x)=1/2*d(u)

3. Rewrite the integral by splitting x3 into x2⋅x and substituting the variables.

(∫_^)(x2*e(x2)⋅x*d(x))=(∫_^)(u*eu⋅1/2*d(u))

4. Factor out the constant from the integral.

1/2*(∫_^)(u*eu*d(u))

5. Apply integration by parts using the formula (∫_^)(w*d(v))=w*v−(∫_^)(v*d(w)) where w=u and d(v)=eu*d(u)

6. Calculate the parts d(w)=d(u) and v=eu

1/2*(u*eu−(∫_^)(eu*d(u)))

7. Evaluate the remaining integral.

1/2*(u*eu−eu)+C

8. Factor out the common term eu

1/2*eu*(u−1)+C

9. Back-substitute u=x2 to return to the original variable.

1/2*e(x2)*(x2−1)+C

Final Answer

(∫_^)(x3*e(x2)*d(x))=1/2*e(x2)*(x2−1)+C

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