University Calculus: Early Transcendentals (3rd Ed)
5.5, 5.6
2.1.1 Substitution and the Chain Rule
Reversing the Chain Rule \(\frac{d}{dx}[f(g(x))]=f’(g(x))g’(x)\)
yields the Substitution Rule \(\int f’(g(x))g’(x)\,dx=f(g(x))+C\).
This is often abbreviated as \(\int f’(u)\,du=f(u)+C\)
by using the substitutions \(u=g(x)\) and \(du=g’(x)dx\).
Example
Find \(\int 4(3+4x)^2\,dx\).
Example
Find \(\int 3u^2\sin(u^3)\,du\).
Example
Find \(\int \frac{x}{x^2+1}\,dx\).
Example
Find \(\int \frac{4\sinh(\ln t)}{t}\,dt\).
2.1.2 Substitution in Definite Integrals
When dealing with definite integrals, you may either convert the
boundaries to \(u\)-values, or you must substitute back for the original
variable before plugging in boundaries.
Example
Compute \(\int_{1/4}^{1/2} 4(3+4x)^2\,dx\).
Example
Compute \(\int_0^1 z\sqrt{1-z}\,dz\).
Example
Compute \(\int_0^{\pi/4}\tan^2\theta\sec^2\theta\,d\theta\).
2.1.3 Antiderivatives of Trigonometric Functions
The antiderivatives of the basic trig functions (besides sine/cosine)
may be derived by using Substitution.
Example
Use Substitution to find \(\int\tan\theta\,d\theta\).
Example
Prove that \(\int\csc x\,dx = -\ln|\csc x+\cot x|+C\).
(Optional)
Prove \(
\int \cos^{n+2} x\,dx
=
\frac{\cos^{n+1} x\sin x}{n+2}+\frac{n+1}{n+2}\int\cos^n x\,dx
\). (Hint: take the derivative of both sides.)
(Optional)
Find \(\int \cos^4 x\,dx\) using the above formula.
University Calculus: Early Transcendentals (3rd Ed)
8.2
2.3.1 Integrating Products of Sine and Cosine
To integrate a function of the form \(\sin^m x\cos^n x\) where
at least one of \(m,n\) is odd, use these identities to
substitute \(u=\sin x,du=\cos x\,dx\) or
\(u=\cos x,du=-\sin x\,dx\).
\(\cos^2 x=1-\sin^2 x\)
\(\sin^2 x=1-\cos^2 x\)
Example
Find \(\int\sin^3\theta\cos^4\theta\,d\theta\).
Example
Find \(\int\sin^2(2y)\cos^5(2y)\,dy\).
If both \(m,n\) are even, then one of these identities must be used:
\(\cos^2 x=\frac{1}{2}+\frac{1}{2}\cos(2x)\)
\(\sin^2 x=\frac{1}{2}-\frac{1}{2}\cos(2x)\)
Example
Find \(\int\cos^2 x\,dx\).
Example
Find \(\int\sin^2 z\cos^2 z\,dz\).
2.3.2 Integrating Products of Secant and Tangent
To integrate a function of the form \(\sec^m x\tan^n x\),
use these identities to
substitute \(u=\tan x,du=\sec^2 x\,dx\) or
\(u=\sec x,du=\sec x\tan x\,dx\).
\(\tan^2 x=\sec^2 x-1\)
\(\sec^2 x=\tan^2 x+1\)
Example
Find \(\int\tan^3\theta\sec^3\theta\,d\theta\).
Example
Find \(\int\sec^4 x\tan^5 x\,dx\).
Exercises for 2.3
Find \(\int\sin^4 x\cos^3 x\,dx\).
Find \(\int\sin^5 \theta\cos^2 \theta\,d\theta\).
Find \(\int\sin^2 x\,dx\).
Find \(\int\cos^4 y\,dy\).
Find \(\int\tan^2 t\sec^4 t\,dt\).
(Optional)
Use integration by parts with cycling to prove
\(
\int\sec^3 x\,dx
=
\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C
\).
(Hint: \(\int\sec x\tan^2 x\,dx=\int\sec x(\sec^2 x-1)\,dx\).)
University Calculus: Early Transcendentals (3rd Ed)
8.3
2.4.1 Substituting for \(a+bx^2\)
To eliminate factors of the form \(a+bx^2\) from an integral,
use the substitution \(a+bx^2=a+a\tan^2\theta=a\sec^2\theta\)
with \(-\pi/2<\theta<\pi/2\).
Example
Find \(\int\frac{z^2}{4+9z^2}\,dz\).
Example
Compute \(\int_0^2\frac{1}{\sqrt{16+4x^2}}\,dx\).
(Recall \(\int\sec\theta\,d\theta=\ln|\sec\theta+\tan\theta|+C\).)
2.4.2 Substituting for \(a-bx^2\)
To eliminate factors of the form \(a-bx^2\) from an integral,
use the substitution \(a-bx^2=a-a\sin^2\theta=a\cos^2\theta\)
with \(-\pi/2\leq\theta\leq\pi/2\).
Note that this is only valid when \(|x|\leq\sqrt{a/b}\),
which is guaranteed when \(a-bx^2\) is under a square root.
Example
Find \(\int(4-25s^2)^{-3/2}\,ds\).
Example
Find \(\int\frac{x^3}{\sqrt{1-4x^2}}\,dx\).
2.4.3 Substituting for \(bx^2-a\)
To eliminate factors of the form \(bx^2-a\) from an integral,
use the substitution \(bx^2-a=a\sec^2\theta-a=a\tan^2\theta\)
with \(0\leq\theta<\pi/2\).
Note that this is only valid when \(x\geq\sqrt{a/b}\),
which will be assumed in our problems.
Example
Prove \(\int\frac{1}{x\sqrt{x^2-1}}\,dx=\inverse\sec x+C\)
where \(x>1\).
Example
Find \(\int\frac{\sqrt{y^2-16}}{y}\,dy\) where \(y\geq 4\).
2.4.4 Using Inverse Trigonometric Antiderivatives
Sometimes, a simpler substiution may be combined with the following
antiderivatives to obtain a solution more elegantly.
\(\int\frac{1}{1+x^2}\,dx=\inverse\tan x+C\).
\(\int\frac{1}{\sqrt{1-x^2}}\,dx=\inverse\sin x + C\).
\(\int\frac{1}{x\sqrt{x^2-1}}\,dx=\inverse\sec x+C\)
where \(x>1\).
Example
Find \(\int\frac{3}{\sqrt{9-x^2}}\,dx\) without using a
trigonometric substitution.
(Note: the quiz given in class had a typo:
\(\int \frac{1}{\sqrt{9-y^2}}\,dy\), making
\(\inverse\sin(\frac{y}{3})+C\) or
“None of the Above” the correct solution. The full solution below
is for the version without a typo.)
(Quiz)
Find \(\int \frac{1}{x\sqrt{4x^2-1}}\,dy\) where \(x>\frac{1}{2}\).
University Calculus: Early Transcendentals (3rd Ed)
8.4
2.5.1 Rational Functions and Partial Fractions
A function of the form \(\frac{f(x)}{g(x)}\) where \(f,g\) are
both polynomials is called rational.
The rational function \(\frac{f(x)}{(x+r)^m}\) may be split into
the partial fractions
\(\frac{A_1}{x+r}+\frac{A_2}{(x+r)^2}+\dots+\frac{A_m}{(x+r)^m}\),
provided the degree of the numerator is less than the denominator.
Example
Expand \(\frac{2x^2-7x+6}{(x-2)^3}\) using partial fractions.
The rational function \(\frac{f(x)}{(x^2+px+q)^n}\) (where
\(x^2+px+q\) is irreducible) may be split into
the partial fractions
\(
\frac{B_1x+C_1}{x^2+px+q}+
\frac{B_2x+C_2}{(x^2+px+q)^2}+
\dots+
\frac{B_mx+C_m}{(x^2+px+q)^n}
\),
provided the degree of the numerator is less than the denominator.
Example
Expand \(\frac{3x^2+2x+4}{x^4+2x^2+1}\) using partial fractions.
When the denominator is a product of \((x+r)^m\) and
\((x^2+px+q)^n\) terms, simply sum up the appropriate
partial fractions for each factor.
Example
Describe the partial fractions which expand the rational function
\(\frac{f(x)}{(x+3)^3(x^2-2x+3)^2}\).
2.5.2 Integrating Partial Fractions
Expanding rational functions using partial fractions allows us
to integrate.
Example Find \(\int\frac{2x^2+5x-9}{(x-1)(x+1)(x-2)}\,dx\).
Example Find \(\int\frac{4y^2+14y+15}{y^3+4y^2+5y}\,dy\).
If the numerator has degree greater than or equal to the denominator,
you will need to use long polynomial division to break down the
rational function first.
Example Find \(\int\frac{2t^3+t^2+3t+2}{(1+t)(1+t^2)}\,dt\).
Exercises for 2.5
Expand \(\frac{4x^2+16x+17}{(x+2)^3}\) using partial fractions.
Expand \(\frac{-y^2+2y-4}{(y^2+4)^2}\) using partial fractions.
Expand \(\frac{3r^3+r^2+3}{r^4+3r^2}\) using partial fractions.
(Quiz)
Which of the following describes the expansion of
\(\frac{f(t)}{(t+1)^2(t^2+9)}\) using partial fractions?
(Assume \(f(t)\) is a polynomial of degree less than 4.)
When encountering an integration problem, it’s useful to spot
certain traits which can identify the best integration method
to apply. The following list isn’t fool-proof, but checking these
in order can help you identify likely techniques for integration.
Use algebra to simplify the integrand first, if possible.
Split up sums into separate integrals as necessary.
Is the integrand a constant multiple of a known derivative?
If so, simply integrate using the constant multiple rule.
Is the integral of the form \(\int cf(g(x))g’(x)\,dx\):
a nested function along with (a constant multiple of)
its derivative?
If so, use integration by substitution with \(u=g(x)\).
Is the integrand a rational function (a fraction of two polynomials)?
If so, try the method of partial fractions
to expand the integrand algebraically.
Does the integrand include only trigonometric functions?
Use trig identities to allow for a direct substitution.
Does the integrand include expressions of the form \(a+bx^2\),
\(a-bx^2\), or \(bx^2-a\)? Use the method of trigonometric
substitution to simplify.
Is the integrand the product of two functions? Integration by
parts may produce a more manageable integral.
At this point, check to make sure you didn’t miss a possibility
above. Otherwise, you may need to use a combination of techniques
from the above to proceed.
Example Find \(\int\sinh x\sqrt{1+\cosh x}\,dx\).
Example Find \(\int 2ze^{3z}\,dz\).
Example Find \(\int\sin^2 \theta+\cos^2 \theta\,d\theta\).
Example Find \(\int\frac{5x^2+12}{x^3+4x}\,dx\).
Example Find \(\int3\sec y\tan y-\frac{1}{1+y^2}\,dy\).