Find the indefinite integra

Question

Find the indefinite integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.)

∫ (sqrt(x)+(8/x)-− 5e^x)dx

Answer 1

This integral can be split into 3 smaller, easier to integrate integrals. I have also rewritten the exponents and pulled constant factors to the outside of the integrals,

∫x^1/2dx + 8 ∫x^-1dx + 5 ∫e^xdx

Now we can take the integral of each part and add them together. Each would technically have it's own +c because they are indefinite, but I will combine those all at the end and continue to call it c. We can do this because it is an arbitrary constant.

∫x^1/2dx = (2/3)*x^3/2 + c

this was found by adding 1 to the exponent and and finding a coefficient to make the antiderivative match the derivative. In this case that was 2/3 which will cancel out to give a coefficient of 1 when we check our work by taking the derivative.

Continuing with the other pieces,

8 ∫x^-1dx = 8*ln|x| +c

**∫1/x dx =ln|x| + c is a good thing to have memorized, google a proof if you're extra curious!

5*∫e^xdx = 5e^x + c

**the derivative of e^x = e^x and is also it's own antiderivative

FINALLY we can add all these together..

∫x^1/2dx + 8 ∫x^-1dx + 5 ∫e^xdx = (2/3)*x^3/2 + c + 8*ln|x| + c + 5e^x + c

= (2/3)*x^3/2 + 8*ln|x| + 5e^x + c

Answer 2

∫ [√x + 8/x - 5e^x]dx = (2/3)x^3/2 + 8ln lxl - 5e^x + C