Computing antiderivatives (integration)

[Nitrate]

Homework Statement

integrate 4e^(2x)^(1/2) - 1/7e^(-pix)
using a guess and check method (haven't learned many rules of integration)

Homework Equations

The Attempt at a Solution

i'm not really sure how to do this integral... i tried
4/(2x)^(1/2)[e^(2x)^(1/2)] using a table of antiderivatives for the first part
but when i differentiated it, it did not give me the original function
i haven't tried the second bit yet.

 

[The1337gamer]

For the second term, do you know how to differentiate exponential functions?

Can you answer these questions, differentiate with respect to x:

e^x
18e^x
4e^2x
e^(x^2)
e^(x^(1/3))
e^(8x^(-2/3))

For the first term you need to use a substitution, try substituting u=x^1/2

 

[HallsofIvy]

If you have $\int f'(x)e^{f(x)}dx$ you could make the substitution $$u= f(x)$$ so that $$du= f'(x)dx$$ and the integral becomes $$\int e^u du= e^u+ C= e^{f(x)}+ C$$

HOWEVER, if that f'(x) is not in the integral originally (and is not a constant), you cannot put it in! Here, I don't believe that $$\int e^{(4x)^{1/2}}dx$$ can be integrated in terms of elementary functions.