Evaluate the Integral

Problem

(∫_^)(t^3*e^(−t^2)*d(t))

Solution

1. Substitute a new variable to simplify the exponent. Let u=t^2 which implies d(u)=2*t*d(t) or t*d(t)=1/2*d(u)

2. Rewrite the integral in terms of u by splitting t^3 into t^2⋅t

(∫_^)(t^2*e^(−t^2)*(t*d(t))=(∫_^)(u*e^(−u)1/2*d(u)))

3. Factor out the constant from the integral.

1/2*(∫_^)(u*e^(−u)*d(u))

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

1/2*(u*(−e^(−u))−(∫_^)(−e^(−u)*d(u)))

5. Evaluate the remaining integral.

1/2*(−u*e^(−u)−e^(−u))+C

6. Factor out the common term −e^(−u)

−1/2*e^(−u)*(u+1)+C

7. Back-substitute u=t^2 to return to the original variable.

−1/2*e^(−t^2)*(t^2+1)+C

Final Answer

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

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