indefinite integrals

Question

Hi Team,

 

Any help with the questions below would be greatly appreciated.

 

f(x)=(3x^5+15xsqrtx-48e^(-2x)+(48/1+x^2))dx

 

Thanks in advance

 

 

Michael

Answer 1

First, we want to write the function in a form that will be easier to integrate.  Any radical terms will be turned into rational exponents.

 

f(x) = ∫(3x⁵ + 15xx^1/2 - 48e^-2x + (48 / (1 + x²)))dx

 

f(x) = ∫(3x⁵ + 15x^3/2 - 48e^-2x + (48 / (1 + x²))dx

 

 

Overall, you will use the power rule for integrals to evaluate the integral.  I think you easily integrate the first two terms.  The last two terms require a different approach.  In the last two terms, you will need to use substitution.  Let call these last two terms sub-integrals.

 

 

For       -48∫e^-2xdx

 

 

Let          u = -2x

             du = -2dx

            (-1/2)du = dx

 

 

Plug in in these variables to get an integral

 

(-48)(-1/2)∫e^udu =

 

24∫e^udu =

 

24e^u =

 

24e^-2x

 

 

 

For           48∫(1 / (1 + x²)))dx

 

 

The anti-derivative of     1/(1+x²)  is   tan^-1(x).

 

48∫(1 / (1 + x²)))dx = 48tan^-1(x)

 

 

 

Then you are going to take the sum of these integrals.  That will be your final integral.  Make sure to add +C at the end.