1116 views

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.

3 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

417 SUBJECT SPECIALISTS

£20 /hr

Degree: Computer Science (Bachelors) - Warwick University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Computing

“Maths and Computer Science lover. With a patient, thorough and friendly teaching style.”

£20 /hr

Degree: Economics and Statistics (Integrated Masters) - St. Andrews University

Subjects offered:Maths, Business Studies+ 1 more

Maths
-Personal Statements-

“I study Economics and Statistics because I enjoy learning about the role mathematics and statistics plays in our everyday.”

MyTutor guarantee

£22 /hr

Degree: Economics (Bachelors) - LSE University

Subjects offered:Maths, Economics+ 1 more

Maths
Economics
-Personal Statements-

“I will provide engaging and rewarding tailored lessons, adapting my teaching to best suit you and do my best to assist you in reaching your full potential”

Currently unavailable: for regular students

Degree: Physics (Masters) - Durham University

Subjects offered:Maths, Physics

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

Do the following vector equations intersect? l = (1 + μ)i + (2 - μ)j + (2μ - 5)k, and m = 2λi + 3j + (2 + λ)k.

y = (x^2)sin(3x). Find dy/dx

Solve the differential equation: dy/dx = tan^3(x)sec^2(x)

Differentiate: y=12x(2x+1)+1/x

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