# Integration help?

• Nov 13th 2012, 07:40 AM
EarlyTravel
Integration help?
How would I go about integrating these two problems?

1. http://i47.tinypic.com/34q2ryq.png dx

2. http://i46.tinypic.com/2wqqtuh.png dx

With #2, I know that Sin integrated is is -Cos x (+c), and that Cos integrated is Sin x (+c), but I'm not sure how to integrate fractions.
• Nov 13th 2012, 07:56 AM
HallsofIvy
Re: Integration help?
Substitutions: In (1) let $u= 1+ x^2$. In (2) let u= 1+ 2sin(x).
• Nov 13th 2012, 07:59 AM
fkf
Re: Integration help?
Quote:

Originally Posted by EarlyTravel
How would I go about integrating these two problems?

1. http://i47.tinypic.com/34q2ryq.png dx

2. http://i46.tinypic.com/2wqqtuh.png dx

With #2, I know that Sin integrated is is -Cos x (+c), and that Cos integrated is Sin x (+c), but I'm not sure how to integrate fractions.

Let
f(x) = 4x/(1+x^2) = 4x(1+x^2)^-1

We're now searching for the primitive function F(x), this is found to be
F(x) = 2*ln(1+x^2)+C

You can try to take the derivative of F(x) and see that it holds. When you write a function and sees ^-1, be aware of the ln(x) function since y = ln(x) => y' = 1/x = x^-1.

The same holds for 2).

Let
f(x) = cos(x)/(1+2sin(x)) = cos(x)(1+2sin(x))^-1

You can now find
F(x) = ln(1+2sin(x))/2+C

Another tip is to start with the ln function, and then se what the derivative is by using the chainrule, then you for example know that in this case you should divide by 2 so it cancel when taking the derivative of the function F(x). But be aware! This only holds when we're talking about constants, since if we would use this strategy on a function, we would have to apply the product rule when calculating the derivative and that would almost surely lead to wrong function.