1. Help with integration!

Hi

I really need help with these two questions, for the first attachement I have no idea how to do that last part circled in pink

For the second attachement(labeled part e.)), I could not get the answer - I got $\displaystyle \frac{4sqrt{2}}{sqrt{3}}arctan(\frac{sqrt{2}}{sqrt {3}}) + c$ but the answer is $\displaystyle \2\frac{sqrt{2}}{sqrt{3}}arctan(\frac{sqrt{2}}{sqr t{3}}) + c$

Thank you so much in advance!

2. Re: Help with integration!

Originally Posted by Tutu
Hi

I really need help with these two questions, for the first attachement I have no idea how to do that last part circled in pink

For the second attachement(labeled part e.)), I could not get the answer - I got $\displaystyle \frac{4sqrt{2}}{sqrt{3}}arctan(\frac{sqrt{2}}{sqrt {3}}) + c$ but the answer is $\displaystyle \2\frac{sqrt{2}}{sqrt{3}}arctan(\frac{sqrt{2}}{sqr t{3}}) + c$

Thank you so much in advance!
1) You have $\displaystyle \int_0^3 f(x) \ dx = 8 = \int_c^d f(x-2) \ dx$

Integration by substitution - the substitution should be pretty easy to spot, bearing in mind the argument of $\displaystyle f(x-2)$.

2) Their answer is correct, I cannot point out what you've done incorrectly unless you post your working OP, likely a small arithmetical error somewhere.

3. Re: Help with integration!

Hi thank you so much!

For the second, attached is my working, I had simply used the formula so its sort of one step..can you help me see where I went wrong?

4. Re: Help with integration!

Originally Posted by Tutu
Hi thank you so much!

For the second, attached is my working, I had simply used the formula so its sort of one step..can you help me see where I went wrong?
You haven't done $\displaystyle \int \frac{dx}{a^2 + (px+q)^2} = \frac{1}{ap} \arctan \left(\frac{px+q}{a}\right) + \mathcal{C}$

but rather:

$\displaystyle \int \frac{dx}{a^2 + (px+q)^2} = \frac{1}{a} \arctan \left(\frac{px+q}{a}\right) + \mathcal{C}$.