Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) ...

Question

Evaluate the integrals in Exercises 33–54.

∫ 2t e^(-t²) dt

Accepted Answer

Welcome back everyone. Evaluate the integral of 4 x e to the power of x 2 dx. We're going to approach this problem using U substitution. Let's suppose that u is equal to x2. Our next step is to find the derivative of u with respect to x. That's the derivative of x 2, which is -2 x. Now, what we can do is simply notice that we have XDX within the integral, right? So let's solve for x multiplied by dx. What we're going to do is multiply both sides by dx and divide both sides by -2. So x multiplied by dx is equal to DU divided by -2, or simply -1/ DU. And then we can rewrite our integrals for multiplied by. XDX basically gives us. 1/22 DU multiplied by e. To the power of x 2, so e to the power of u. Now let's combine the constants, that would be -2, and we can factor out -2 and integrate e to the power of uDU, which is an elementary integral. Its value is e to the power of u plus a constant of integration c. So multiplying by -2, we get 2 e to the power of u plus c. And finally, replacing you with X2, we are going to get -2e to the power of X2 + C. And this is our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 33–54.∫ 2t e^(-t²) dt

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Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt