Fundamental Integration formulas

Question

1. Evaluate

∫ 3x / 1-3x^{2 dx}

using change of a variable.

2.Derive the following formula

∫sec x dx = 1/2 In(1+sinx/1-sinx)+C

3.Derive a formula for ∫xⁿ In x dx where

a. n≠-1 n=-1

4.Derive a formula for ∫tan x secⁿ x dx where n≠0.

Answer 1

I will give you some hints..but you need to work on these for yourself.

1. Make the substitution u=3x²
2. Show that (1+sin x)/(1-sin x)=(sec x + tan x)²...that is a pretty easy identity to prove; start by multiplying the numerator and denominator by (1+sin x).
3. Make the substitution u=ln x
4. Make the substitution u = sec x, then du=sec x tan x.