Substitution Integral Problem Help!!!!?

Question

i am trying to study for a test and these 4 problems got me worried. they are even numbers from my book so i cant even look the answer up in the back of the book and work backwards. please help and explain what you did. thank you!

1. ∫(x+3)/((x^2)+6x+1)dx

2. ∫(x^3)/((1-x^8)^(1/2))dx

3. ∫(sec^(2)(x))*(e^tanx)dx

4. ∫x(1-2x)^(1/2)dx

if you can help with any of them please do. thank you!

Answer 1

#1 is a u substitution problem.

Set u = x^2 + 6x + 1
du = (2x + 6) dx
Divide (2x + 6) by 2, which gives you x+3

This will then give you a 1/2 on the outside of the integral. The integral then becomes this...

(1/2) * integral ( du / u )

You should recognize this as a ln. The answer is:

( ln ( x^2 + 6x + 1) ) / 2

#2
You should recognize this as an inverse trig function. Sine in particular.

The general form for the inverse sine in an integral looks like 1/sqrt(1-u^2)

Here, your u is x^4. So the answer is:

invsin(x^4)/4

#3
We know that the derivative of tan(x) is sec^2(x) so this simple becomes e^(tan(x)).

#4
Not really sure.

Answer 2

Do a u-substitution for u=a million+z^3. From there you will see that du = 3z^2 dz and z^2 dz = a million/3 du. this might show you how to rewrite the crucial as : a million/3 * int(u^(-a million/3) du) which will desire to be somewhat common to clean up.