Integration by parts

What you will be able to do

• use the formula for integration by parts and choose $u$ well;
• integrate $\ln x$ and $\tan^{-1}x$, and use parts twice or in a cycle;
• recognise the form $e^x\bigl[f(x) + f'(x)\bigr]$.

The product rule, read backwards

The product rule says $(uv)' = u'v + uv'$. Rearrange it as $uv' = (uv)' - u'v$ and integrate both sides:

This is integration by parts. It is useful when $\int u'v\,dx$ is easier than the integral you started with. In short form, with $du = u'\,dx$ and $dv = v'\,dx$, it reads $\int u\,dv = uv - \int v\,du$.

Choosing $u$

Pick $u$ from the first of these that appears in the integrand. The word that helps many students is ILATE.

1. Inverse trigonometric functions, such as $\tan^{-1}x$.
2. Logarithms, such as $\ln x$.
3. Algebraic terms, such as $x$ or $x^2$.
4. Trigonometric functions, such as $\sin x$.
5. Exponentials, such as $e^x$.

The idea is that $u$ should get simpler when differentiated, and $dv$ should be easy to integrate. Some books use LIATE, with the same meaning.

Worked examples

Example 1. Find $\displaystyle\int x\,e^x\,dx$.

1. Take $u = x$ and $dv = e^x\,dx$. Then $du = dx$ and $v = e^x$.
2. The formula gives $x\,e^x - \int e^x\,dx = x\,e^x - e^x + C$.

Example 2. Find $\displaystyle\int \ln x\,dx$. There seems to be only one factor. Write it as $\ln x \cdot 1$.

1. Take $u = \ln x$ and $dv = dx$. Then $du = \dfrac{dx}{x}$ and $v = x$.
2. The formula gives $x\ln x - \int x \cdot \dfrac{1}{x}\,dx = x\ln x - x + C$.

Example 3. Find $\displaystyle\int \tan^{-1}x\,dx$.

1. Take $u = \tan^{-1}x$ and $dv = dx$. Then $du = \dfrac{dx}{1 + x^2}$ and $v = x$.
2. The formula gives $x\tan^{-1}x - \int \dfrac{x}{1 + x^2}\,dx$.
3. The last integral is $\dfrac{1}{2}\ln(1 + x^2)$ by substitution. The answer is $x\tan^{-1}x - \dfrac{1}{2}\ln(1 + x^2) + C$.

Example 4 (twice). Find $\displaystyle\int x^2 e^x\,dx$.

1. Take $u = x^2$ and $dv = e^x\,dx$. The formula gives $x^2 e^x - \int 2x\,e^x\,dx$.
2. Use Example 1 for the last integral: $\int 2x\,e^x\,dx = 2x\,e^x - 2e^x$.
3. The answer is $e^x\left(x^2 - 2x + 2\right) + C$.

Example 5 (a cycle). Find $\displaystyle I = \int e^x\sin x\,dx$.

1. Take $u = \sin x$ and $dv = e^x\,dx$. Then $I = e^x\sin x - \int e^x\cos x\,dx$.
2. Use parts again on the last integral, with $u = \cos x$. It gives $\int e^x\cos x\,dx = e^x\cos x + \int e^x\sin x\,dx = e^x\cos x + I$.
3. So $I = e^x\sin x - e^x\cos x - I$, which gives $2I = e^x(\sin x - \cos x)$.
4. The answer is $I = \dfrac{e^x}{2}(\sin x - \cos x) + C$.

The shortcut $e^x\bigl[f(x) + f'(x)\bigr]$

Differentiate $e^x f(x)$ by the product rule. You get $e^x f(x) + e^x f'(x) = e^x\bigl[f(x) + f'(x)\bigr]$. Read it backwards:

Example 6. Find $\displaystyle\int e^x\left(\dfrac{1}{x} - \dfrac{1}{x^2}\right)dx$. Here $f(x) = \dfrac{1}{x}$ and $f'(x) = -\dfrac{1}{x^2}$. So the integral is $\dfrac{e^x}{x} + C$, with no parts needed.

Practise with new numbers

Try it in Python

The formula for $\int x\,e^{kx}\,dx$ came from integration by parts. The program checks it for three values of $k$ by taking the slope of the answer.

Common mistakes

• Choosing $u$ the wrong way round. If the second integral is harder than the first, swap the roles of $u$ and $dv$.
• Losing a sign. The formula has a minus: $uv - \int v\,du$.
• Forgetting the factor in $v$. For $dv = e^{kx}\,dx$ we have $v = \dfrac{e^{kx}}{k}$, not $e^{kx}$.
• Stopping at the first step. After the first step, look at the new integral. It may need parts again.
• Using parts on a product that is only substitution. $\int x\,e^{x^2}\,dx$ is a substitution.

Remember these

• $\int u\,dv = uv - \int v\,du$, from the product rule.
• ILATE: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. Choose $u$ from the earlier one.
• $\ln x$ and $\tan^{-1}x$ are integrated with $dv = dx$.
• $\int e^x\bigl[f(x) + f'(x)\bigr]\,dx = e^x f(x) + C$.

Check yourself

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

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

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

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

4. Find $\displaystyle\int e^x\left(\tan x + \sec^2 x\right)dx$.

5. Find $\displaystyle\int e^{2x}\cos x\,dx$.

References

• OpenStax, Calculus Volume 2, Chapter 3, section 3.1 Integration by Parts.
• NCERT, Mathematics Part II for Class XII, Chapter 7, the section on integration by parts.