Integration by Parts
Integration by Parts is a powerful technique for integrating products of functions.
It is based on the product rule for derivatives, reversed for integrals.
Where Integration by Parts Comes From
Start with the product rule:
$$\frac{d}{dx}(uv) = u^\prime(v)+(u)v^\prime$$
Integrate both sides:
$$uv = \int (u^\prime(v)+{u}v^\prime) dx =\int u^\prime(v) dx + \int (u)v^\prime dx$$
Rearrange:
$$\int (u)v^\prime dx = uv – \int u^\prime(v) dx$$
Integration by Parts Formula
we need to set:
-
u: a function you differentiate
-
dv: a function you integrate
or
we need to set:
-
$$f(x)$$: a function you differentiate
-
$$g^\prime(x)$$: a function you integrate
When Do We Use Integration by Parts?
Use it when the integrand is a product of two types of functions:
Common situations:
Polynomial × Exponential
$$\int x\cdot e^{x} dx$$
Polynomial × Trig
$$\int x\cdot\sin(x) dx$$
Logarithm or Inverse trig
$$\int \ln(x) dx$$
Reduce powers
$$\int x^{n}\cdot e^{x} dx , \int x^{n}\sin{x} dx$$
Choosing u and dv
Logarithmic Polynomial(monomial) Trigonometry Exponent
Derivative $$\longleftarrow$$ $$\longrightarrow$$ Integral
Example:
$$\int x\cdot e^{x}dx$$
-
x: monomial (algebraic)
-
$$e{x}$$: exponential
→ choose $$u=x , dv=e^{x}$$ or $$f(x)=x , g^\prime(x)=e^{x}$$
Example 1:
$$\int x\cdot e^{x}dx$$
set:
$$u=x \rightarrow du=dx / dv=e^{x} \rightarrow v= e^{x}$$
apply the formula:
$$\int x\cdot e^{x}dx = x\cdot e^{x}-\int1\cdot e^{x}dx = x\cdot e^{x}-e^{x}+C$$
Example 2:
$$\int x\sin(x) dx$$
set:
$$u=x \rightarrow du=dx / dv=\sin(x) \rightarrow v=-\cos(x)$$
apply the formula:
$$\int x\sin(x) dx=-x\cos(x)+\int1\cot\cos(x)dx=-x\cos(x)+\sin(x)+C$$
Example 3: (Logarithm Integration)
$$\int\ln(x)dx$$
set:
$$u=\ln(x) \rightarrow du=\frac{1}{x} / dv=1 \rightarrow v=x$$
apply the formula:
$$\int\ln(x)dx=x\cdot\ln(x)-\int x\cdot\frac{1}{x}dx =x\cdot\ln(x)-\int 1 dx = x\cdot\ln(x)-x+C$$
Example 4:
$$\int e^{x}\cos(x)dx$$
set:
$$u=\cos(x) \rightarrow du=-\sin(x) / dv=e^{x} \rightarrow v=e^{x}$$
apply the formula:
$$e^{x}\cos{x}+\int e^{x}\sin(x)dx$$ and then set again:
$$u=\sin(x) \rightarrow du=\cos(x) / dv=e^{x} \rightarrow v=e^{x}$$
apply again the formula:
$$e^{x}\cos{x}+e^{x}\sin(x)-\int e^{x}\cos(x)dx$$ and here is important, we need to transposition $$-\int e^{x}\cos(x)dx$$ then
$$2\cdot\int e^{x}\cos(x)dx=e^{x}\cos{x}+e^{x}\sin(x)$$ and divide by 2 both side
$$\int e^{x}\cos(x)dx=\frac{e^{x}\cos{x}+e^{x}\sin(x)}{2}$$