Integrals of trigonometric functions

Integral calculus, grade 11 (5 units) Lesson 12 of 20Learning

Integrals of trigonometric functions

Three basic integrals

In the differential calculus course you learnt, in radians: $\displaystyle {(\sin x)'}={\cos x}$ , $\displaystyle {(\cos x)'}={-\sin x}$ , $\displaystyle {(\tan x)'}={\frac{1}{\cos^2x}}$ . Each of these derivatives, read backwards, gives an integral:

\displaystyle \int\cos x\,dx=\sin x+C

\displaystyle \int\sin x\,dx=-\cos x+C

\displaystyle \int\frac{1}{\cos^{2}x}\,dx=\tan x+C

Watch the sign: $\displaystyle {\int\sin x\,dx}={-\cos x}$ , because $\displaystyle {(-\cos x)'}={\sin x}$ .
The formulas hold only when the angle is in radians: in degrees the derivatives are multiplied by $\displaystyle \frac{\pi}{180}$ .

A linear argument

The rule $\displaystyle {\int f(mx+b)\,dx}={\frac{1}{m}F(mx+b)+C}$ works for trigonometric functions too. In the examples below:

$\displaystyle {\int\cos3x\,dx}={\frac{1}{3}\sin3x+C}$ .
$\displaystyle {\int\sin\left(2x-\frac{\pi}{3}\right)dx}={-\frac{1}{2}\cos\left(2x-\frac{\pi}{3}\right)+C}$ .
$\displaystyle {\int\frac{1}{\cos^2\frac{x}{2}}\,dx}={2\tan\frac{x}{2}+C}$ : here $\displaystyle {m}={\frac{1}{2}}$ , and dividing by $\displaystyle \frac{1}{2}$ is multiplying by 2.

An inner derivative

The rule $\displaystyle {\int\left(u(x)\right)^n u'(x)\,dx}={\frac{\left(u(x)\right)^{n+1}}{n+1}+C}$ works here too. The derivative of $\displaystyle \sin x$ is $\displaystyle \cos x$ , so:

$\displaystyle {\int\sin^3x\cos x\,dx}={\frac{\sin^4x}{4}+C}$ ( $\displaystyle {u}={\sin x}$ ).
$\displaystyle {\int\frac{\sin x}{\cos^2x}\,dx}={-\int(\cos x)^{-2}\cdot(-\sin x)\,dx}={-\frac{(\cos x)^{-1}}{-1}+C}={\frac{1}{\cos x}+C}$ ( $\displaystyle {u}={\cos x}$ ).

The area of one arch

The example below calculates the area between one arch of $\displaystyle {y}={\sin x}$ and the $\displaystyle x$ -axis, between $\displaystyle {x}={0}$ and $\displaystyle {x}={\pi}$ :

\int_0^\pi\sin x\,dx=\left[-\cos x\right]_0^\pi=-\cos\pi-(-\cos0)=1+1=2

\displaystyle f(x)=\sin x

Over a whole period, from 0 to $\displaystyle 2\pi$ , the lower arch equals the upper one, and the integral is 0: $\displaystyle {\left[-\cos x\right]_0^{2\pi}}={-1+1}={0}$ .
So to calculate the area of several arches, calculate each arch separately, or multiply the area of one.

Exercise 1

Find the integrals.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Calculate the integrals.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The function $\displaystyle {f(x)}={\sin4x}$ is given.

Mark the points where the graph meets the $\displaystyle x$ -axis.

\displaystyle f(x)=\sin4x

Calculate the shaded area, between the graph and the $\displaystyle x$ -axis in the first arch to the right of the origin.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the area between the graph and the $\displaystyle x$ -axis for $\displaystyle 0\le x\le\pi$ ? Explain without calculating another integral.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: integral calculus, grade 11 (5 units)

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Basic integrals

The indefinite integral – all the antiderivatives
$\displaystyle \int f(x)\,dx=F(x)+C$ $\displaystyle F^{\prime}(x)=f(x)$
A power
$\displaystyle \int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
A constant
$\displaystyle \int k\,dx=kx+C$
$\displaystyle \int\frac{1}{x^{2}}\,dx=-\frac{1}{x}+C$
One over a root
$\displaystyle \int\frac{1}{\sqrt{x}}\,dx=2\sqrt{x}+C$
A root
$\displaystyle \int\sqrt{x}\,dx=\frac{2}{3}x\sqrt{x}+C$

The rules of integration

A constant factor
$\displaystyle \int k\cdot f(x)\,dx$ $\displaystyle =$ $\displaystyle k\int f(x)\,dx$
A sum and a difference
$\displaystyle \int\left(f(x)\pm g(x)\right)dx$ $\displaystyle =$ $\displaystyle \int f(x)\,dx\pm\int g(x)\,dx$
A linear inner function
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
A composite function times the inner derivative
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
A power of a function times its derivative
$\displaystyle \int\left(u(x)\right)^{n}\cdot u^{\prime}(x)\,dx=\frac{\left(u(x)\right)^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$

Trigonometric functions (in radians)

$\displaystyle \int\cos x\,dx=\sin x+C$
$\displaystyle \int\sin x\,dx=-\cos x+C$
$\displaystyle \int\frac{1}{\cos^{2}x}\,dx=\tan x+C$
From $\displaystyle {\cos2x}={1-2\sin^2x}$
$\displaystyle \sin^{2}x$ $\displaystyle =$ $\displaystyle \frac{1-\cos2x}{2}$
From $\displaystyle {\cos2x}={2\cos^2x-1}$
$\displaystyle \cos^{2}x$ $\displaystyle =$ $\displaystyle \frac{1+\cos2x}{2}$
From $\displaystyle {\sin2x}={2\sin x\cos x}$
$\displaystyle \sin x\cos x$ $\displaystyle =$ $\displaystyle \frac{1}{2}\sin2x$
From $\displaystyle {\sin^2x+\cos^2x}={1}$
$\displaystyle \tan^{2}x$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^{2}x}-1$

The accumulation function and the definite integral

The accumulation function of $\displaystyle f$ from $\displaystyle a$
$\displaystyle F_{a}(x)=\int_{a}^{x}f(t)\,dt$
The fundamental theorem
$\displaystyle F_{a}^{\prime}(x)=f(x)$
The definite integral, $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The same limit
$\displaystyle \int_{a}^{a}f(x)\,dx=0$
Swapping the limits
$\displaystyle \int_{b}^{a}f(x)\,dx$ $\displaystyle =$ $\displaystyle -\int_{a}^{b}f(x)\,dx$
Joining intervals
$\displaystyle \int_{a}^{b}f(x)\,dx+\int_{b}^{c}f(x)\,dx$ $\displaystyle =$ $\displaystyle \int_{a}^{c}f(x)\,dx$

Areas and accumulation

An area above the $\displaystyle x$ -axis
$\displaystyle S=\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\geq0$
An area below the $\displaystyle x$ -axis
$\displaystyle S=-\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\leq0$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The displacement from the velocity
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$