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:



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.


this simplifies to:


This is the final answer for this question.


Michael H. GCSE Maths tutor, A Level Maths tutor, GCSE Physics tutor,...

2 years ago

Answered by Michael, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Viktoria B.

Degree: BEng Aerospace Engineering with Integrated Foundation Year (Bachelors) - Manchester University

Subjects offered:Maths, Russian+ 2 more

Further Mathematics

“My personal motto is: Practice not until you get it right, but until you cannot get it wrong.”

MyTutor guarantee

£20 /hr

Francesco S.

Degree: Theoretical Physics (Bachelors) - Edinburgh University

Subjects offered:Maths, Physics+ 2 more

Further Mathematics

“Sometimes exams can look a bit scary, but have no fear! I am Theoretical Physics student with a long experience in tutoring and I am here to help! ”

MyTutor guarantee

£20 /hr

Caroline H.

Degree: Chemistry (Masters) - Edinburgh University

Subjects offered:Maths, Chemistry+ 1 more


“I am a chemistry student at Edinburgh University. Although I have achieved highly I have always had to put in a little more effort than most going through material just that extra time. I am keen to be able to offer others the support ...”

About the author

Michael H.

Currently unavailable: for regular students

Degree: Physics (Masters) - Durham University

Subjects offered:Maths, Physics


“Hello, my name's Michael, and I'm a first year physicist at Durham. I've always been passionate about science and maths, leading to my degree choice, and I'd like to share my experience in achieving top grades in these subjects.”

You may also like...

Posts by Michael

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

Show that the orbital period of a satellite is given by T^2=(4pi^2r^3)/(GM) where r is the orbital radius, G is the gravitational constant and M is the mass of the Earth. Then find the orbital radius of a geostationary satellite.

Other A Level Maths questions

Find dy/dx when y = 2ln(2e-x)

How do you find the gradient of a parametric equation at a certain point?

What are volumes of revolution and how are they calculated?

Find f'(x) and f''(x) when f(x) = 3x^2 +7x - 3

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss