Calculus 5.5 U Substitution

Question

I'm having trouble solving these U substitution problems. I still want to solve the main questions on my own so I'll give samples instead. If you could help me solve these I think I can get a grasp on the main questions.∫

∫(7e^x + 5) / (e^x)

(e^x) / (7e^x + 5)

Solve using U substitution, as they're indefinite integrals I know each one ends with +C.

Answer 1

You wouldn't typically use a u-sub for the first problem - in my perspective it makes it harder.

∫(7e^x + 5)/e^x dx = ∫7e^x/e^x dx + ∫5/e^x dx = ∫7 dx + ∫5e^-x dx = 7x - 5e^-x + C

In the second example, let u = e^x, then du/dx = e^x or rewriting it, dx = du/e^x or dx = du/u

So the integral becomes:

∫u/(7u + 5) du/u = ∫1/(7u+5) du = (1/7)ln(7u + 5) + C = (1/7)ln(7e^x + 5)