Integration by Parts

Definition

Integration by parts is a formula used to integrate the product of two functions.

If u=u(x) and v=v(x), the following equation holds:

uvdx=uvuvdx,

where u=dudx and v=dvdx.

The formula for integration by parts requires one of the functions appearing in the integrand to be differentiated, and the other to be integrated. To apply this formula to a product of two functions, choose one of the functions to be the function that is differentiated and label it u. The other function is to be integrated, and is therefore labelled v.

Note: On some occasions the integrand appearing in the term uvdx may still be a product of two functions of x. In this case the integration by parts formula would need to be applied again to compute uvdx.

Some care is required when making the choices for u and v. The function to be differentiated, u, should be a function that will be reduced to a simpler function after differentiating. Polynomial functions are a good choice for u, since repeated differentiation of a polynomial will always produce a constant. Trigonometric functions such as cosx are not good choices for u as they retain their trigonometric form after differentiation, and are therefore better choices for v.

The formula can also be applied to definite integration between limits a and b:

abuvdx=[uv]ababuvdx.

Derivation of the Formula

Suppose there are two functions u=u(x) and v=v(x). Then by the product rule, the derivative of their product is

ddx[uv]=udvdx+vdudx.

Integrate both sides with respect to x:

ddx[uv]dx=udvdxdx+vdudxdx.

By the fundamental theorem of calculus,

ddx[uv]dx=uv,

hence:

uv=udvdxdx+vdudxdx.

Rearranging this gives the formula for integration by parts:

udvdxdx=uvvdudxdx.

Worked Example

Example 1

Use integration by parts to find 0π/2(1+x)sinxdx.

Solution

Recall the formula for integration by parts:

abuvdx=[uv]ababuvdx.

The first step is to make suitable choices for u and v. The two functions appearing in the integrand are 1+x and sinx. The choice for u should be a function that will become simpler when differentiated; here sinx would be a bad choice, as the derivative of sinx is just another trigonometric function, cosx. The derivative of 1+x, however, is a constant, so we choose u=1+x.

The choices for u and v are u=1+x and v=sinx.

It can be convenient to lay out the expressions for u, v and their derivatives in the following way:

u=1+x,v=???,u=???,v=sinx.

The gaps for u and v can be filled in after performing the relevant calculations.

Differentiating u with respect to x gives u:

\[u' = \frac{\mathrm{d </div>{\mathrm{d} x}\bigl[1+x\bigr] = 1,\]

and integrating v with respect to x gives v:

v=sinxdx=cosx.

Note: It is not necessary to include the constant of integration when computing v.

The expressions for u, v, u and v are:

u=1+x,v=cosx,u=1,v=sinx.

Substituting these into the formula for integration by parts gives:

0π/2(1+x)sinxdx=[(1+x)(cosx)]0π/20π/21(cosx)dx=[(1+x)cosx]0π/2+0π/2cosxdx.

First evaluate [(1+x)cosx]0π/2

[(1+x)cosx]0π/2=(1+π2)cosπ2{(1+0)cos0}=(1+π2)0+11=1.

Substitute this back in the formula to obtain:

0π/2(1+x)sinxdx=1+0π/2cosxdx=1+[sinx]0π/2=1+(sinπ2sin0)=1+(10)=2.

The value of the integral is 0π/2(1+x)sinxdx=2.

Note: Suppose different choices were made for u and v. Let u=sinx and v=(1+x). Then u=cosx and v=x+x22. Substituting these expressions into the formula for integration by parts yields: 0π/2(1+x)sinxdx=[(x+x22)sinx]0π/20π/2(x+x22)cosxdx. It is immediately obvious that the integral on the right hand side is more complicated than the original problem, thus emphasising the importance of making appropriate choices for u(x) and v(x).

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Video Examples

Example 1

Prof. Robin Johnson uses integration by parts to find 0πx2sinxdx.

Example 2

Prof. Robin Johnson uses integration by parts to find 0πetsintdt.

Example 3

Prof. Robin Johnson uses integration by parts to find (1+2x)exdx.

Example 4

Prof. Robin Johnson uses integration by parts to find xsin(12x)dx.

Example 5

Prof. Robin Johnson uses integration by parts to find xarctanxdx.

Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

Test Yourself

Test yourself: Numbas test on integration by parts

External Resources

Whiteboard maths

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