Integration by Substitution

Integration by Substitution

There is a theorem that will help you with substitution for integration. It is called Change of Variables for Definite Integrals.

what the theorem looks like is this

$\int _{a}^{b}f(x)\operatorname {d} x=\int _{\alpha }^{\beta }f(g(u))g\prime (u)\operatorname {d} u$

In order to get $\alpha$ you must plug a into the function g and to get $\beta$ you must plug b into the function g.

The tricky part is trying to identify what you want to make your u to be. Some times substitution will not be enough and you will have to use the rules for integration by parts. That will be covered in a different section

Steps

$\int \limits _{x=a}^{x=b}f(x)dx$	$=\int \limits _{x=a}^{x=b}f(x)\ {\frac {du}{du}}\ dx$	(1)	i.e. ${\frac {du}{du}}=1$
	$=\int \limits _{x=a}^{x=b}{\left(f(x)\ {\frac {dx}{du}}\right)\left({\frac {du}{dx}}\right)}\ dx$	(2)	i.e. ${\frac {dx}{du}}\cdot {\frac {du}{dx}}=1$
	$=\int \limits _{x=a}^{x=b}\left(f(x)\ {\frac {dx}{du}}\right)g'(x)\ dx$	(3)	i.e. ${\frac {du}{dx}}=g'(x)$
	$=\int \limits _{x=a}^{x=b}h(g(x))g'(x)dx$	(4)	i.e. Now equate $\left(f(x)\ {\frac {dx}{du}}\right)$ with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(g(x))}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\limits_{x=a}^{x=b}h(u)g'(x)dx}	(5)	i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=u}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\limits_{u=g(a)}^{u=g(b)}h(u)du}	(6)	i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=\frac{du}{dx}dx=g'(x)dx}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\limits_{u=c}^{u=d}h(u)du}	(7)	i.e. We have achieved our desired result

Example 1

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{2} x(x^2+1)^2 \operatorname {d}x}

Instead of making this a big polynomial we will just use the substitution method.

Step 1

Identify your u

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = x^2+1}

Step 2

Identify Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname {d}u}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname {d}u = 2x\operatorname {d}x}

Step 3

Now we plug in our limits of integration to our u to find our new limits of integration

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0, u =0^2 + 1 = 1}

and when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 2, u = 2^2 + 1 = 5}

Now our integration problem looks something like this

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {1}{2} \int_{0}^{5} (x^2 + 1)^2 (2x)\operatorname {d}x}

Step 4

write your new integration problem

When we plug in our u it looks like

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {1}{2} \int_{0}^{5} (u)^2 \operatorname {d}u}

Step 5

Evaluate the Integral

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {1}{2} \left[\frac {1}{3} u^3 \right]_{0}^{5}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {1}{2} \left[\left(\frac {1}{3} * 5^3 \right) - \left(\frac {1}{3} * 0^3 \right)\right]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {1}{2} \left[\frac {1}{3} * 125 \right]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {1}{2} \left[\frac {125}{3}\right]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {125}{6}}

As you can see this all simplified fairly nice. Using substitution will be hard, for most people, at first. Once you get the hang of doing this it should come to you faster and faster each time.

Example 2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 3x^2(x^3+1)^5dx}

we see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3x^2} is the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3+1} . Letting

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=x^3+1}

we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{du}{dx}=3x^2}

or, in order to apply it to the integral,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=3x^2dx}

With this we may write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 3x^2(x^3+1)^5dx=\int u^5du=\frac{u^6}{6}+C=\frac{(x^3+1)^6}{6}+C}

Note that it was not necessary that we had exactly the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} in our integrand. It would have been sufficient to have any constant multiple of the derivative.

For instance, to treat the integral

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^4\sin(x^5)dx}

we may let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=x^5} . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=5x^4dx}

and so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{du}{5}=x^4dx}

the right-hand side of which is a factor of our integrand. Thus,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^4\sin(x^5)dx=\int\frac{\sin(u)}{5}du=-\frac{\cos(u)}{5}+C=-\frac{\cos(x^5)}{5}+C}

Resources

• Recognizing Derivatives and Substitution Rules, WikiBooks: Calculus
• Integration by Substitution, WikiBooks: High School Calculus
• Example 1. Produced by Professor Zachary Sharon, UTSA

• Example 2. Produced by TA Catherine Sporer, UTSA

Indefinite Integrals Using Substitution

• Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Integration by Substitution Part 1 by James Sousa, Math is Power 4U
• Integration by Substitution Part 2 by James Sousa, Math is Power 4U
• Ex 1: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 2: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 3: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 4: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 5: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 6: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 7: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 8: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 9: Indefinite Integration Using Substitution by James Sousa, Math is Power 4U

• Integration using U-Substitution by patrickJMT
• U-Substitution - More Complicated Examples by patrickJMT

• U-Substitution Example 1 by Krista King
• U-Substitution Example 2 by Krista King
• U-Substitution Example 3 by Krista King
• U-Substitution Example 4 by Krista King
• U-Substitution Example 5 by Krista King
• U-Substitution Example 6 by Krista King
• U-Substitution Example 7 by Krista King

• How To Integrate Using U-Substitution by The Organic Chemistry Tutor

Definite Integrals Using Substitution

• Definite Integration Using Subsitution by James Sousa, Math is Power 4U
• Ex 1: Definite Integration Using Substitution - Change Limits of Integration? by James Sousa, Math is Power 4U
• Ex 2: Definite Integration Using Substitution - Change Limits of Integration? by James Sousa, Math is Power 4U
• Ex 1: Definite Integration Using Substitution by James Sousa, Math is Power 4U
• Ex 2: Definite Integration Using Substitution by James Sousa, Math is Power 4U

• Integration by U-Substitution, Definite Integral by patrickJMT
• U-Substitution: When Do I Have to Change the Limits of Integration ? by patrickJMT

• U-Substitution Integration, Indefinite & Definite Integral by The Organic Chemistry Tutor

Licensing

Content obtained and/or adapted from:

• Recognizing Derivatives and Substitution Rules, WikiBooks: Calculus under a CC BY-SA license

• Integration by Substitution, WikiBooks: High School Calculus under a CC BY-SA license