Evaluate the integrals in Exercises 53–76.
73. ∫(from 0 ...

Question

Evaluate the integrals in Exercises 53–76.

73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))

Accepted Answer

Welcome back everyone. Evaluate the integral from 0 up to 11 of 2 of e to the power of x dx divided by 1 + 4 e to the power of 2 x. For this problem, let's go ahead and rewrite the integral using the properties of exponents that's integral from 0 to ln of 2 of e to the power of x. D x divided by 1 + 2 e to the power of x squared, right, and this allows us to use the u substitution. Let's suppose that u is equal to 2 e to the power of x. And now we can identify the derivative of e with respect to x, which is the derivative of 2e the power of x, and that's basically 2e the power of x. We can now solve for e to the power of x dx because that's what we have in the numerator, and specifically we can multiply both sides by dx. And divided both sides by 2, which gives us one half the u. Well then, and now let's go ahead and change the limits of integration. The lower 1 u1 is going to be 2e, to the power of 0, which is 2. And the upper 1, U2 is going to be 2 e to the power of eon of 2, which is 2 multiplied by 2, and that's 4. So now we can rewrite the integral in terms of u and DU. So that's integral from 2 to 4. In the numerator we have 1/2 d u or simply d u because we can factor out 1/2 as a constant. In the denominator we have 1 + u 2. Notice that this is an elementary integral. We get 1 half followed by inverse of tangent of u. Evaluated from 2 to 4. And now we can calculate the result. That's 1/2 in parent inverse of tangent of 4 minus inverse of tangent of 2. What we can do additionally here is simplify the expression. Using The formulas from trigonometry. So first of all, we're going to rewrite the fraction as. Inverse of tangent of where. Minus inverse of tangent of 2 divided by 2. And now we can use the formula. Tangents of a minus b equals Tangents Of a minus tangent of b Divided by One plus. Tangent a multiplied by tangent B. Now let's suppose that a is equal to inverse of tangent of 4. And B is. In verse of the ancient of 2. So now if we apply the formula, we get tangent of A, which is going to be 4 minus tangent B, which is going to be 2. Divided by 1 plus the product of the 2, so that's 1 + 4 multiplied by 2, and we get 2 divided by 9. So, A minus B. Is going to be equal to. In verse of tangent. Of 2 divided by 9. Well done, so now we have simplified it. Let's label our integral with i, and the integral i is going to be equal to 1/2. Notice that we have our constant. In verse of tangent of 2 divided by 9, that's our final answer, and thank you for watching.

Evaluate the integrals in Exercises 53–76.73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))

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Evaluate the integrals in Exercises 53–76.
73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))