Integration by Parts

Reference > Calculus: Integration

Description

A method of integration that transforms the integral of a product of two functions using the following:

∫u dv=uv−∫v du\int u \, dv=uv-\int v \, d u
The goal is to transform the integral into another form that is easier to solve. It is the inverse of the product rule in differentiation.


Examples

Example 1

∫xex dx\int x{e}^{x} \, dx
1
Use Integration by Parts on ∫xex dx\int x{e}^{x} \, dx.
Let u=xu=x, dv=exdv={e}^{x}, du=dxdu=dx, v=exv={e}^{x}

2
Substitute the above into uv−∫v duuv-\int v \, du.
xex−∫ex dxx{e}^{x}-\int {e}^{x} \, dx

3
The integral of ex{e}^{x} is ex{e}^{x}.
xex−exx{e}^{x}-{e}^{x}

4
Add constant.
xex−ex+Cx{e}^{x}-{e}^{x}+C

Done


 

Example 2

∫ln⁡xx5 dx\int \frac{\ln{x}}{{x}^{5}} \, dx
1
Use Integration by Parts on ∫ln⁡xx5 dx\int \frac{\ln{x}}{{x}^{5}} \, dx.
Let u=ln⁡xu=\ln{x}, dv=1x5dv=\frac{1}{{x}^{5}}, du=1x dxdu=\frac{1}{x} \, dx, v=−14x4v=-\frac{1}{4{x}^{4}}

2
Substitute the above into uv−∫v duuv-\int v \, du.
−ln⁡x4x4−∫−14x5 dx-\frac{\ln{x}}{4{x}^{4}}-\int -\frac{1}{4{x}^{5}} \, dx

3
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
−ln⁡x4x4+14∫1x5 dx-\frac{\ln{x}}{4{x}^{4}}+\frac{1}{4}\int \frac{1}{{x}^{5}} \, dx

4
Use Power Rule: ∫xn dx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
−ln⁡x4x4−116x4-\frac{\ln{x}}{4{x}^{4}}-\frac{1}{16{x}^{4}}

5
Add constant.
−ln⁡x4x4−116x4+C-\frac{\ln{x}}{4{x}^{4}}-\frac{1}{16{x}^{4}}+C

Done


 

Example 3

∫xcos⁡(3x) dx\int x\cos{(3x)} \, dx
1
Use Integration by Parts on ∫xcos⁡3x dx\int x\cos{3x} \, dx.
Let u=xu=x, dv=cos⁡3xdv=\cos{3x}, du=dxdu=dx, v=sin⁡3x3v=\frac{\sin{3x}}{3}

2
Substitute the above into uv−∫v duuv-\int v \, du.
xsin⁡3x3−∫sin⁡3x3 dx\frac{x\sin{3x}}{3}-\int \frac{\sin{3x}}{3} \, dx

3
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
xsin⁡3x3−13∫sin⁡3x dx\frac{x\sin{3x}}{3}-\frac{1}{3}\int \sin{3x} \, dx

4
Use Integration by Substitution on ∫sin⁡3x dx\int \sin{3x} \, dx.
Let u=3xu=3x, du=3 dxdu=3 \, dx, then dx=13 dudx=\frac{1}{3} \, du

5
Using uu and dudu above, rewrite ∫sin⁡3x dx\int \sin{3x} \, dx.
∫sin⁡u3 du\int \frac{\sin{u}}{3} \, du

6
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
13∫sin⁡u du\frac{1}{3}\int \sin{u} \, du

7
Use Trigonometric Integration: the integral of sin⁡u\sin{u} is −cos⁡u-\cos{u}.
−cos⁡u3-\frac{\cos{u}}{3}

8
Substitute u=3xu=3x back into the original integral.
−cos⁡3x3-\frac{\cos{3x}}{3}

9
Rewrite the integral with the completed substitution.
xsin⁡3x3+cos⁡3x9\frac{x\sin{3x}}{3}+\frac{\cos{3x}}{9}

10
Add constant.
xsin⁡3x3+cos⁡3x9+C\frac{x\sin{3x}}{3}+\frac{\cos{3x}}{9}+C

Done


 
See Also

Cymath foriOSAndroid