Integration by substitution

What you will be able to do

• recognise an inner function and its derivative inside an integrand;
• carry out a substitution in clear steps and return to $x$ at the end;
• integrate $\tan x$, $\cot x$, $\sec x$ and $\operatorname{cosec} x$.

The chain rule, read backwards

The chain rule says $\dfrac{d}{dx}F(g(x)) = F'(g(x))\,g'(x)$. Read it from right to left. If $F' = f$, then

The method is a change of letter. Put $u = g(x)$. Then $du = g'(x)\,dx$, and the integral becomes $\int f(u)\,du$. This is easier, because the inner function has gone.

The steps

1. Choose $u$. Pick the inner function, often the part inside brackets, under a root, or in a power.
2. Find $du$. Differentiate: $du = g'(x)\,dx$.
3. Replace. Write the whole integral in $u$. No $x$ may remain. A missing constant factor is fine, and you fix it with a number in front.
4. Integrate in $u$.
5. Return to $x$. Put $u = g(x)$ back.

A shortcut for a linear inside

When the inner function is $ax + b$, we have $du = a\,dx$. So

The factor $\dfrac{1}{a}$ is easy to forget. The simulation shows why it is needed.

Worked examples

Example 1. Find $\displaystyle\int 2x\cos(x^2)\,dx$.

1. Let $u = x^2$. Then $du = 2x\,dx$.
2. The integral becomes $\int \cos u\,du = \sin u + C$.
3. Return to $x$: $\sin(x^2) + C$.

Example 2. Find $\displaystyle\int x^2 e^{x^3}\,dx$.

1. Let $u = x^3$. Then $du = 3x^2\,dx$, so $x^2\,dx = \tfrac{1}{3}du$.
2. The integral becomes $\tfrac{1}{3}\int e^u\,du = \tfrac{1}{3}e^u + C$.
3. Return to $x$: $\tfrac{1}{3}e^{x^3} + C$.

Example 3. Find $\displaystyle\int \dfrac{x}{x^2 + 1}\,dx$.

1. Let $u = x^2 + 1$. Then $x\,dx = \tfrac{1}{2}du$.
2. The integral becomes $\tfrac{1}{2}\int \dfrac{du}{u} = \tfrac{1}{2}\ln|u| + C$.
3. Since $x^2 + 1 > 0$, the answer is $\tfrac{1}{2}\ln(x^2 + 1) + C$.

Example 4. Find $\displaystyle\int x\sqrt{x + 2}\,dx$. Here the derivative of the inside is $1$, so the extra $x$ is not a derivative. Replace it too.

1. Let $u = x + 2$. Then $x = u - 2$ and $dx = du$.
2. The integral becomes $\int (u - 2)\sqrt{u}\,du = \int \left(u^{3/2} - 2u^{1/2}\right)du = \tfrac{2}{5}u^{5/2} - \tfrac{4}{3}u^{3/2} + C$.
3. Return to $x$: $\tfrac{2}{5}(x + 2)^{5/2} - \tfrac{4}{3}(x + 2)^{3/2} + C$.

The family $f'/f$

When the top is the derivative of the bottom, put $u = f(x)$. Then $du = f'(x)\,dx$, and

This gives four integrals that every student should know.

• $\displaystyle\int \tan x\,dx = \int \dfrac{\sin x}{\cos x}\,dx = -\ln|\cos x| + C = \ln|\sec x| + C$.
• $\displaystyle\int \cot x\,dx = \int \dfrac{\cos x}{\sin x}\,dx = \ln|\sin x| + C$.
• $\displaystyle\int \sec x\,dx = \ln|\sec x + \tan x| + C$.
• $\displaystyle\int \operatorname{cosec} x\,dx = \ln|\operatorname{cosec} x - \cot x| + C$.

For the third one, multiply the top and the bottom by $\sec x + \tan x$. The top becomes $\sec^2 x + \sec x\tan x$, which is the derivative of the bottom $\sec x + \tan x$. The fourth works the same way with $\operatorname{cosec} x - \cot x$.

Practise with new numbers

Try it in Python

Each row of the program has an integrand and the answer that a substitution gave. The program checks every answer by taking its slope. The last row has a deliberate slip.

Common mistakes

• Forgetting the constant factor. From $du = 2x\,dx$ we get $x\,dx = \tfrac{1}{2}du$. Leaving out the half gives an answer twice too large.
• Mixing $x$ and $u$. Once you write $u$, every $x$ must go. Return to $x$ only at the end.
• Stopping at $u$. The question is in $x$, so the answer must be too.
• Forcing a substitution. If the derivative of the inside is not there, try expanding, or replace $x$ as well, as in Example 4.
• Dropping the absolute value. $\ln|f(x)|$ is the safe form when $f(x)$ can be negative.

Remember these

• Substitution reverses the chain rule: $u = g(x)$ and $du = g'(x)\,dx$.
• For a linear inside, $\int f(ax + b)\,dx = \dfrac{1}{a}F(ax + b) + C$.
• $\int \dfrac{f'}{f}\,dx = \ln|f| + C$.
• $\int \tan x\,dx = \ln|\sec x|$, $\int \cot x\,dx = \ln|\sin x|$, $\int \sec x\,dx = \ln|\sec x + \tan x|$ and $\int \operatorname{cosec} x\,dx = \ln|\operatorname{cosec} x - \cot x|$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find $\displaystyle\int x\sin(x^2)\,dx$.

2. Find $\displaystyle\int e^{3x + 2}\,dx$.

3. Find $\displaystyle\int \dfrac{\ln x}{x}\,dx$ for $x > 0$.

4. Find $\displaystyle\int \dfrac{\sin x}{1 + \cos x}\,dx$.

5. Find $\displaystyle\int \dfrac{dx}{x\ln x}$ for $x > 1$.

References

• OpenStax, Calculus Volume 1, Chapter 5, section 5.5 Substitution.
• NCERT, Mathematics Part II for Class XII, Chapter 7, the section on integration by substitution.