Use any method to evaluate the integrals in Exercises 55–66.

Question

∫ 2^x / (2²x + 2^x - 2) dx

Accepted Answer

Welcome back everyone. Evaluate the integral 3 to the power of x dx divided by 32 x minus 3 x minus 6. We're going to start this problem by introducing a u substitution. Let's suppose that u is equals to 3 to the power of x, and then the derivative of u with respect to x is going to be 3 to the power of x multiplied by ln of 3, right? So we can now show that dx. Or specifically what we can do is notice that our integral contains 3 to the power of x, d x, and the numerator, so we can just rearrange and show that 3 to the power of x multiplied by d x. is equal to duu divided by ln of 3 or simply 1 divided by ln of 3 multiplied by duu. So we can now rewrite the integral in terms of you and the. The numerator becomes 1 divided by eon of 3DU, and we can factor out the constant. In the denominator we have u2 minus u minus 6. And now let's perform factorization. So we get 1 divided by eon of 3 integral of d u divided by. You minus 3, you plus 2. So what we're going to do from here is perform partial fraction decomposition. Let's rewrite our integrant function, that will be 1 divided by u minus 3 u + 2. And we can basically split it into a divided by the first linear factor u minus 3. Plus b divided by the 2nd linear factoru + 2. Now let's multiply both sides by u minus 3 U plus 2, which gives us 1 on the left hand side. On the right hand side, we get a multiplied by u plus 2. Plus b multiplied by u minus 3. Let's start By identifying a. And to find a, we're going to set u equals 3 to eliminate the b term. Which gives us 1 equals a multiplied by 3 + 2, which is 5, right? So A is going to be 1 divided by 5 or 1/5. Now let's set u equals -2 to eliminate a. And -2 minus 3 is -5, so we get 1 equals -5B and therefore B is -1/5. Well done, so we can now rewrite the integral as follows. We have our constant 1 divided by eon of 3. Integral A divided by U minus 3 is going to be 1/5 divided by U minus 3, or basically 1/5 multiplied by 1 divided by U minus 3. Pless B which is -15. Multiplied by 1 divided by U + 2. The EU We can now factor out 1/5 to get 1 divided by. 5 Ellen of 3. Integral 1 divided by U minus 3 minus 1 divided by U + 2. These are now elementary integrals, and we can go ahead and integrate so we get one divided by. 5 ln of 3 in parenthes. Alan Of the absolute value. Of U minus 3. Minus ln of the absolute value of u + 2. Plus a constant of integration c. Now let's use the properties of logarithms to simplify the answer. That's going to be 1 divided by 5 ln of 3. Multiplied by ln of the absolute value of u minus 3 divided by u plus 2. Plessy constant of integration c. And now let's replace you with 3 to the power of x. Which essentially gives us the final answer for this problem. Thank you for watching.

Use any method to evaluate the integrals in Exercises 55–66.∫ 2^x / (2²x + 2^x - 2) dx

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Use any method to evaluate the integrals in Exercises 55–66.
∫ 2^x / (2²x + 2^x - 2) dx