# Integration by Substitution/Definite Integral

## Theorem

Let $\phi$ be a real function which has a derivative on the closed interval $\closedint a b$.

Let $I$ be an open interval which contains the image of $\closedint a b$ under $\phi$.

Let $f$ be a real function which is continuous on $I$.

If $\map \phi a \le \map \phi b$, then the definite integral of $f$ from $a$ to $b$ can be evaluated by:

- $\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$

where $t = \map \phi u$.

If $\map \phi a > \map \phi b$, then the definite integral of $f$ from $a$ to $b$ can be evaluated by:

- $\ds - \int_{\map \phi b}^{\map \phi a} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$

The technique of solving an integral in this manner is called **integration by substitution**.

### Corollary

Let $f : \R \to \R$ be a real function.

Let $f$ be integrable.

Let $a$, $b$, and $c$ be real numbers.

Then:

- $\ds \int_{a - c}^{b - c} \map f t \rd t = \int_a^b \map f {t - c} \rd t$

## Proof

Let $F$ be an antiderivative of $f$.

We have:

\(\ds \map {\frac \d {\d u} } {\map F t}\) | \(=\) | \(\ds \map {\frac \d {\d u} } {\map F {\map \phi u} }\) | Definition of $\map \phi u$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac \d {\d t} \map F {\map \phi u} \dfrac \d {\d u} \map \phi u\) | Chain Rule for Derivatives | |||||||||||

\(\ds \) | \(=\) | \(\ds \map f {\map \phi u} \dfrac \d {\d u} \map \phi u\) | as $\map F t = \ds \int \map f t \rd t$ |

Hence $\map F {\map \phi u}$ is an antiderivative of $\map f {\map \phi u} \dfrac \d {\d u} \map \phi u$.

Thus:

\(\ds \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u\) | \(=\) | \(\ds \bigintlimits {\map F {\map \phi u} } a b\) | Fundamental Theorem of Calculus: Second Part | |||||||||||

\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \map F {\map \phi b} - \map F {\map \phi a}\) |

If $\map \phi a \le \map \phi b$, we also have:

\(\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t\) | \(=\) | \(\ds \bigintlimits {\map F t} {\map \phi a} {\map \phi b}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map F {\map \phi b} - \map F {\map \phi a}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u\) | from $(1)$ |

which was to be proved.

If instead $\map \phi a > \map \phi b$, we have:

\(\ds - \int_{\map \phi b}^{\map \phi a} \map f t \rd t\) | \(=\) | \(\ds - \bigintlimits {\map F t} {\map \phi b} {\map \phi a}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map F {\map \phi b} - \map F {\map \phi a}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u\) | from $(1)$ |

which was to be proved.

$\blacksquare$

## Also known as

Because the most usual substitution variable used is $u$, this method is often referred to **Integration by Substitution** as **$u$-substitution** in the source works for introductory-level calculus courses.

Some sources refer to this technique as **Change of Variable**, but that has a number of different meanings depending upon context.

## Proof Technique

The usefulness of the technique of Integration by Substitution stems from the fact that it may be possible to choose $\phi$ such that $\map f {\map \phi u} \dfrac \d {\d u} \map \phi u$ (despite its seeming complexity in this context) may be easier to integrate.

If $\phi$ is a trigonometric function, the use of trigonometric identities to simplify the integrand is called **integration by trigonometric substitution** (or simply **trig substitution**).

Care must be taken to address the domain and image of $\phi$.

This consideration frequently arises when inverse trigonometric functions are involved.

## Sources

- 1944: R.P. Gillespie:
*Integration*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable - 1976: K. Weltner and W.J. Weber:
*Mathematics for Engineers and Scientists*... (previous) ... (next): $6$. Integral Calculus: Appendix: Rules and Techniques of Integration: $2.2$*Integration by substitution* - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 13.22$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**change of variable** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**change of variable**

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- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 4.5, \S 8.4$