Evaluate the Integral integral of te^(t^2) with respect to t

Problem

(∫_^)(t*e(t2)*d(t))

Solution

1. Identify the method of integration by noticing that the exponent t2 has a derivative 2*t which is a multiple of the factor t outside the exponential.

2. Substitute u=t2 to simplify the integrand.

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

4. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(eu1/2*d(u))

5. Factor out the constant 1/2 from the integral.

1/2*(∫_^)(eu*d(u))

6. Integrate the exponential function with respect to u

1/2*eu+C

7. Back-substitute the original variable by replacing u with t2

1/2*e(t2)+C

Final Answer

(∫_^)(t*e(t2)*d(t))=1/2*e(t2)+C

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