Integrals by trigonometric identities

Integral calculus, grade 11 (5 units) Lesson 13 of 20Learning

Integrals by trigonometric identities

An identity before integrating

$\displaystyle \sin^2x$ , $\displaystyle \cos^2x$ and $\displaystyle \tan^2x$ have no basic integral, and no inner derivative in them. A trigonometric identity turns them into a sum of functions that have integrals. The double-angle identities are on the exam’s formula sheet: $\displaystyle {\cos2x}={2\cos^2x-1}={1-2\sin^2x}$ and $\displaystyle {\sin2x}={2\sin x\cos x}$ . From them:

\displaystyle \sin^{2}x

\displaystyle =

\displaystyle \frac{1-\cos2x}{2}

\displaystyle \cos^{2}x

\displaystyle =

\displaystyle \frac{1+\cos2x}{2}

\displaystyle \sin x\cos x

\displaystyle =

\displaystyle \frac{1}{2}\sin2x

\displaystyle \tan^{2}x

\displaystyle =

\displaystyle \frac{1}{\cos^{2}x}-1

The last one comes from dividing $\displaystyle {\sin^2x+\cos^2x}={1}$ by $\displaystyle \cos^2x$ .

Three examples

In the examples below, replace each expression by an identity, and then find the integral:

$\displaystyle {\int\sin^2x\,dx}={\int\left(\frac{1}{2}-\frac{1}{2}\cos2x\right)dx}={\frac{x}{2}-\frac{\sin2x}{4}+C}$ .
$\displaystyle {\int\tan^2x\,dx}={\int\left(\frac{1}{\cos^2x}-1\right)dx}={\tan x-x+C}$ .
$\displaystyle {\int\left(\sin x+\cos x\right)^2dx}={\int\left(\sin^2x+2\sin x\cos x+\cos^2x\right)dx}={\int\left(1+\sin2x\right)dx}={x-\frac{\cos2x}{2}+C}$ .

The mean of

\displaystyle \sin^2x

The example below calculates $\displaystyle \int_0^\pi\sin^2x\,dx$ :

\int_0^\pi\sin^2x\,dx=\left[\frac{x}{2}-\frac{\sin2x}{4}\right]_0^\pi=\frac{\pi}{2}-0=\frac{\pi}{2}

$\displaystyle {\sin^2x+\cos^2x}={1}$ , so $\displaystyle {\int_0^\pi\left(\sin^2x+\cos^2x\right)dx}={\pi}$ .
Over half a period, $\displaystyle \sin^2x$ and $\displaystyle \cos^2x$ accumulate the same area: the graph of one is the graph of the other shifted by $\displaystyle \frac{\pi}{2}$ . So each of them is $\displaystyle \frac{\pi}{2}$ .

\displaystyle f(x)=\sin^{2}x

\displaystyle g(x)=\cos^{2}x

Exercise 1

Find the integrals with an identity.

/ – 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

Danny calculated $\displaystyle \int_0^{2\pi}\left(\left(2+\sin x\right)^2-\left(2-\sin x\right)^2\right)dx$ , and got 0.
Simplify the expression inside the integral, and explain why the result is 0.

/ – 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
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$