Evaluate the integrals in Exercises 69–134. The integrals are ...

Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (e^x + e^(3x)) / e^(2x) dx

Accepted Answer

Welcome back everyone. Evaluate the the integral e to the power of x plus e to the power of 5x divided by e to the power of 3 xDX. For this problem, let's first of all analyze the function that we're integrating e to the power of x plus e to the power of 5 x divided by e to the power of 3 x and let's perform division. That's e to the power of x divided by e to the power of 3 x. Plus e to the power of 5 x divided by e to the power of 3 x which is E to the power of x minus 3x, that's -2 x plus. E to the power of 5 x minus 3x, which is e to the power of 2 x. Now, notice that here we have a similar form to. Cash, right, hyperbolic cosine. So what we're going to do is remember that cache of x is equal to e to the power of x plus e to the power of neg x divided by 2. Notice that if we multiply both sides by 2, we will have 2 cos of x is equal to each the power of x plus e to the power of negative x divided by 2. And in this case, the main difference is that instead of X, we have two eggs, right? So we can now show that e to the power of 2 x plus e to the power of -2x, which we got. Is going to be equal to 2. Cash off 2 X. So what we're integrating is. To cash. 2 x d x and we can factor out the constant to get to integral of cash 2 x d x. And now we are ready to integrate. We can use U substitution. Let's suppose that U is 2x. And the derivative of u with respect to x is going to be the derivative of 2x, which is 2. So DX is going to be DU divided by 2 or simply 1 half DU. The x is equal to 1/2 d u, and this means we can rewrite the integrals 2 integral of cash of u. D x becomes 1/2 d u we can factor out 1/2 and simply write the u with ND and pick run. Notice that 2 divided by 2 is 1, so we are simply integrating cash UDU. The integral of cash is cinch, so we get cinch of u plus a constant of integration c. And finally, we're going to replace you with 2 eggs to get. Cinch of 2 x plus c, which is our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.∫ (e^x + e^(3x)) / e^(2x) dx

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (e^x + e^(3x)) / e^(2x) dx