Evaluate the integrals in Exercises 87–96.
91. ∫₁^(√2) x...

Question

Evaluate the integrals in Exercises 87–96.

91. ∫₁^(√2) x·2^(x²) dx

Accepted Answer

Welcome back everyone. Evaluate the definite integral from 0 to 1 of 3 to the power of x 2 multiplied by x dx. Aus 1 divided by ln of 3, B 2 divided by ln of 3, C 1 divided by 2 ln of 3, and D 3 divided by 2 ln of 3. We're going to approach this problem using U substitution. Let's suppose that U is equal to x2. We can then show that the derivative of u with respect to x is the derivative of x 2, which is 2 x. And since we have XDX in the integral, we're going to rearrange for x multiplied by dx. We can multiply both sides by DX and divide both sides by 2. So we get DU divided by 2, which is 1 half DU. From here we are going to change the limits of integration. The lower one is going to be 0 squad, which is 0. And the upper 1, U2 is going to be 1 squared, which is 1. So the elements of integration remain the same from 0 to 1. We have 3 to the power of x2, which is 3 of U. Followed by x dx, which is one half d u, right, we can factor out one half and simply write d u within the integrant. Now we are ready to integrate because this is an elementary integral and its value is 3 to the power of u divided by ln of 3. We want to evaluate the result from 0 to 1. And we can do that by applying the fundamental theorem of calculus part 2. Let's also factor out the constant, which is 1 divided by 2 of 3. And in parentheses we're going to take 31 minus 30. We get 3 minus 1, which is 2. In the numerator. And we're dividing by 2 of 3. We can cancel out the 2, and we're going to get 1 divided by eon of 3 which corresponds to the answer of choice a. Thank you for watching.

Evaluate the integrals in Exercises 87–96.91. ∫₁^(√2) x·2^(x²) dx

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Evaluate the integrals in Exercises 87–96.
91. ∫₁^(√2) x·2^(x²) dx