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

Problem

(∫_^)(x^3*e^(x^2)*d(x))

Solution

1. Substitute a new variable to simplify the exponent by letting u=x^2

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 x^3 into x^2⋅x and substituting the variables.

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

4. Factor out the constant from the integral.

1/2*(∫_^)(u*e^u*d(u))

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

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

1/2*(u*e^u−(∫_^)(e^u*d(u)))

7. Evaluate the remaining integral.

1/2*(u*e^u−e^u)+C

8. Factor out the common term e^u

1/2*e^u*(u−1)+C

9. Back-substitute u=x^2 to return to the original variable.

1/2*e^(x^2)*(x^2−1)+C

Final Answer

(∫_^)(x^3*e^(x^2)*d(x))=1/2*e^(x^2)*(x^2−1)+C

Want more problems? Check here!