Integration by Parts

Continuing on the path of reversing derivative rules in order to make them useful for integration, we reverse the product rule.

Integration by parts

If $y=uv$ where $u$ and $v$ are functions of $x$ , then

$y'=(uv)'=u'v+uv'$

Rearranging,

$uv'=(uv)'-u'v$

Therefore,

$\int uv'dx=\int (uv)'\ dx-\int u'v\ dx$

Therefore,

$\int uv'\ dx=uv-\int vu'\ dx$

or

$\int u\ dv=uv-\int v\ du$

This is the integration by parts formula. It is very useful in many integrals involving products of functions, as well as others.

For instance, to treat

$\int x\sin(x)dx$

we choose $u=x$ and $dv=\sin(x)dx$ . With these choices, we have $du=dx$ and $v=-\cos(x)$ , and we have

$\int x\sin(x)dx=-x\cos(x)-\int {\big (}-\cos(x){\big )}dx=-x\cos(x)+\int \cos(x)dx=\sin(x)-x\cos(x)+C$

Note that the choice of $u$ and $dv$ was critical. Had we chosen the reverse, so that $u=\sin(x)$ and $dv=x\ dx$ , the result would have been

${\frac {x^{2}\sin(x)}{2}}-\int {\frac {x^{2}\cos(x)}{2}}dx$

The resulting integral is no easier to work with than the original; we might say that this application of integration by parts took us in the wrong direction.

So the choice is important. One general guideline to help us make that choice is, if possible, to choose $u$ to be the factor of the integrand which becomes simpler when we differentiate it. In the last example, we see that $\sin(x)$ does not become simpler when we differentiate it: $\cos(x)$ is no simpler than $\sin(x)$ .

An important feature of the integration by parts method is that we often need to apply it more than once. For instance, to integrate

$\int x^{2}e^{x}dx$

we start by choosing 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} and 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 dv=e^xdx} to get

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^2e^xdx=x^2e^x-2\int xe^xdx}

Note that we still have an integral to take care of, and we do this by applying integration by parts again, 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 u=x} and 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 dv=e^xdx} , which gives us

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^2e^xdx=x^2e^x-2\int xe^xdx=x^2e^x-2(xe^x-e^x)+C=x^2e^x-2xe^x+2e^x+C}

So, two applications of integration by parts were necessary, owing to the power 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^2} in the integrand.

Note that any power of x does become simpler when we differentiate it, so when we see an integral of the form

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^nf(x)dx}

one of our first thoughts ought to be to consider using integration by parts 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 u=x^n} . Of course, in order for it to work, we need to be able to write down an antiderivative for 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 dv} .

Example

Use integration by parts to 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 \int e^x\sin(x)dx}

Solution: If we 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=\sin(x)} and 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 v'=e^xdx} , then 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 u'=\cos(x)dx} and 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 v=e^x} . Using our rule for integration by parts gives

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 e^x\sin(x)dx=e^x\sin(x)-\int e^x\cos(x)dx}

We do not seem to have made much progress.

But if we integrate by parts again 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 u=\cos(x)} and 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 v'=e^xdx} and hence 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'=-\sin(x)dx} and 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 v=e^x} , we obtain

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 e^x\sin(x)dx=e^x\sin(x)-e^x\cos(x)-\int e^x\sin(x)dx}

We may solve this identity to find the anti-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 \sin(x)e^x} and obtain

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 e^x\sin(x)dx=\frac{e^x\big(\sin(x)-\cos(x)\big)}{2}+C}

With definite integral

For definite integrals the rule is essentially the same, as long as we keep the endpoints.

Integration by parts for definite integrals Suppose f and g are differentiable and their derivatives are continuous. 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 \int\limits_a^b f(x)g'(x)dx=\big(f(x)g(x)\big)\bigg|_a^b-\int\limits_a^b f'(x)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 =f(b)g(b)-f(a)g(a)-\int\limits_a^b f'(x)g(x)dx} .

This can also be expressed in Leibniz notation.

$\int \limits _{a}^{b}u\ dv=(uv){\Big |}_{a}^{b}-\int \limits _{a}^{b}v\ du.$

More Examples

Examples Set 1: Integration by Parts

Exercises

Evaluate the following using integration by parts.

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 -4\ln(x)dx}
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 (38-7x)\cos(x)dx}
3. 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_0^\tfrac{\pi}{2} (-6x+45)\cos(x)dx}
4. 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(5x+1)(x-6)^4 dx}
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 \int\limits_{-1}^1 (2x+8)^3(2-x)dx}

Exercise Solutions

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 4x-4x\ln(x)+C}
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 (38-7x)\sin(x)-7\cos(x)+C}
3. 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 51-3\pi}
4. 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{(5x+1)(x-6)^5}{5}-\frac{(x-6)^6}{6}+C}
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 1916.8}

Resources

• Videos and exercises on integration by parts at Khan Academy

Videos

Integration by Parts: The Basics by James Sousa

Integration by Parts (After Integration by Parts Basics) by James Sousa

Integration by Parts - Additional Examples by James Sousa

Integration by Parts: The Basics by James Sousa

Deriving the Integration by Parts Formula - Easy! by patrickJMT

Integration by Parts Made Easy! by patrickJMT

Integration by Parts - Indefinite Integral by patrickJMT

Integration By Parts - Using IBP's Twice by patrickJMT

Integration by Parts - A Loopy Example! by patrickJMT

Integration by Parts - Definite Integral by patrickJMT

How does integration by parts work? by Krista King

Integration by Parts by Krista King

Integration by Parts Example 2 by Krista King

Integration by Parts Example 3 by Krista King

Integration by Parts Example 4 by Krista King