How do I integrate terms with sin^2(x) and cos^2(x) in them? For example integrate (1+sin(x))^2 with respect to x

As you will be aware, it is not easy to directly integrate terms involving sin^2(x) and cos^2(x), so we use a substitution to turn them into something which we can integrate.

From the double angle formula for cosine we have:

cos^2(x)=(cos(2x)+1)/2

sin^2(x)=(1-cos(2x))/2

This substitution removes the terms of sin^2(x) and cos^2(x) which we can't integrate and replaces them with terms involving cos(2x) which integrates to sin(2x)/2, which we know from integration by inspection.

Now consider the example above, integrate (1+sin(x))^2:

To start this question we need to expand the brackets so that we can integrate each term individually. (1+sin(x))^2 expands to 1+2sin(x)+sin^2(x).

Now we have three separate terms which we can integrate:

1 integrates to x

2sin(x) integrates to -2cos(x)

Now substitute sin^2(x) for 1/2-cos(2x)/2. This integrates to x/2-sin(2x)/4

To get our final answer we now add all of the terms we've just integrated together. Remember to include the constant of integration c.

x-2cos(x)+x/2-sin(2x)/4+c

this simplifies to:

3x/2-2cos(x)-sin(2x)/4+c

This is the final answer for this question.