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

$$\int u dv = uv – \int v du$$​

we need to set:

• u: a function you differentiate

• dv: a function you integrate

or

$$\int f(x)g^\prime(x) dx = f(x)g(x) – \int f^\prime(x)g(x) dx$$​

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

$$\int e^{x}\cos(x)dx=\frac{e^{x}\cos{x}+e^{x}\sin(x)}{2}$$