Integrating Trig Functions

How do I integrate sin, cos and sec^2?

The integrals for sine and cosine are

$\int \sin x d x = - \cos x + c$

$\int \cos x d x = \sin x + c$

where $c$ is the constant of integration

Also, from the derivative of $\tan x$

$\int \sec^{2} x d x = \tan x + c$

For the linear function $a x + b$ , where $a$ and $b$ are constants,

$\int \sin (a x + b) d x = - \frac{1}{a} \cos (a x + b) + c$

$\int \cos (a x + b) d x = \frac{1}{a} \sin (a x + b) + c$

$\int \sec^{2} (a x + b) d x = \frac{1}{a} \tan (a x + b) + c$

For calculus with trigonometric functions angles must be measured in radians

Examiner Tip

Remember to add 'c', the constant of integration, for any indefinite integrals

Worked example

Find, in the form

f (x) + c

, an expression for each integral

\int \cos x d x

ii.

\int \sec^{2} (3 x - \frac{π}{3}) d x

i. This is a result you should be able to recall from memory.

\int \cos x d x = \sin x + c

ii. Use the standard result:

\int \sec^{2} (a x + b) d x = \frac{1}{a} \tan (a x + b) + c .

\int \sec^{2} (3 x - \frac{π}{3}) d x = \frac{1}{3} \tan (3 x - \frac{π}{3}) + c

A curve has the following equation:

y = \int 2 \sin (2 x + \frac{π}{6}) d x

The curve passes through the point with coordinates

(\frac{π}{3}, \sqrt{3}) .

Find an expression for

y .

Factor out the 2 in front of the sin first, and then use the standard result:

\int \sin (a x + b) d x = - \frac{1}{a} \cos (a x + b) + c .

y = 2 \int \sin (2 x + \frac{π}{6}) d x y = 2 (- \frac{1}{2} \cos (2 x + \frac{π}{6}) + c)

When multiplying the arbitrary constant

c

, by 2, it is still just an arbitrary constant and so it is still just written as

c .

y = - \cos (2 x + \frac{π}{6}) + c

Substitute in the given coordinate, and evaluate, to find

c .

x = \frac{π}{3}, y = \sqrt{3}

\sqrt{3} = - \cos (\frac{2 π}{3} + \frac{π}{6}) + c \sqrt{3} = - \cos (\frac{5 π}{6}) + c \sqrt{3} = \frac{\sqrt{3}}{2} + c c = \frac{\sqrt{3}}{2}

Rewrite the full expression for

y .

y = - \cos (2 x + \frac{π}{6}) + \frac{\sqrt{3}}{2}

Integrating Trig Functions (Cambridge O Level Additional Maths)

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How do I integrate sin, cos and sec^2?

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