Integration by parts is a technique used to integrate the product of two or more functions. In this textbook chapter, we will discuss integration by parts as it pertains to integrating the product of two functions.
If the integrand is the product of a function $u = u(x)$ and the derivative of another function $dv = v'(x) \, dx$, then:
$$\int u \,dv = uv - \int v \,du$$Integration by parts problems can be solved using an approach called the tabular method. The set-up for the tabular method is shown in each example.
Integration by parts can be broken down into six different approaches depending on how the integrand appears.
Example: $\int x^2\cos(x) \,dx$
Set up a three column table containing alternating + and − signs, the derivatives of the power function down to 0, and the antiderivatives of the trigonometric function up to the same point.
| Sign | D | I |
|---|---|---|
| $+$ | $x^2$ | $\cos(x)$ |
| $-$ | $2x$ | $\sin(x)$ |
| $+$ | $2$ | $-\cos(x)$ |
| $-$ | $0$ | $-\sin(x)$ |
The product of the diagonals of the table is equal to $uv$, while the product of the rows of the table is equal to $v \, du$.
$\int x^2\cos(x) \,dx = x^2\sin(x) + 2x\cos(x) - 2\sin(x) + \int 0\sin(x) \,dx + C$
The term preceding the constant of integration can be dropped because it always evaluates to zero.
$\int x^2\cos(x) \,dx = x^2\sin(x) + 2x\cos(x) - 2\sin(x) + C$
Example: $\int x^{2}e^{x} \,dx$
Set up a three column table containing alternating + and − signs, the derivatives of the power function down to 0, and the antiderivatives of the exponential function up to the same point.
| Sign | D | I |
|---|---|---|
| $+$ | $x^2$ | $e^x$ |
| $-$ | $2x$ | $e^x$ |
| $+$ | $2$ | $e^x$ |
| $-$ | $0$ | $e^x$ |
The product of the diagonals of the table is equal to $uv$, while the product of the rows of the table is equal to $v \, du$.
$\int x^{2}e^{x} \,dx = x^{2}e^{x} - 2xe^x + 2e^x - \int 0 \, e^x \,dx + C$
The term preceding the constant of integration can be dropped because it always evaluates to zero.
$\int x^{2}e^{x} \,dx = x^{2}e^{x} - 2xe^x + 2e^x + C$
Example: $\int x^{2}\ln(x) \,dx$
Set up a three column table containing alternating + and − signs, the derivatives of the logarithmic function, and the antiderivatives of the power function until the product of a row can be integrated without using integration by parts.
| Sign | D | I |
|---|---|---|
| $+$ | $\ln(x)$ | $x^2$ |
| $-$ | $\frac{1}{x}$ | $\frac{1}{3} x^3$ |
The product of the diagonals of the table is equal to $uv$, while the product of the rows of the table is equal to $v \, du$.
$\int x^{2}\ln(x) \,dx = \frac{1}{3} x^{3}\ln(x) - \frac{1}{3} \int x^2 \,dx + C$
Evaluating the integral term gives the final result.
$\int x^{2}\ln(x) \,dx = \frac{1}{3} x^{3}\ln(x) - \frac{1}{9} x^3 + C$
Example: $\int \tan^{-1}(x) \,dx$
Set up a three column table containing alternating + and − signs, the derivatives of the inverse trigonometric function, and the antiderivatives of unity until the product of a row can be integrated without using integration by parts.
| Sign | D | I |
|---|---|---|
| $+$ | $\tan^{-1}(x)$ | $1$ |
| $-$ | $\frac{1}{x^2 + 1}$ | $x$ |
The product of the diagonals of the table is equal to $uv$, while the product of the rows of the table is equal to $v \, du$.
$\int \tan^{-1}(x) \,dx = x\tan^{-1}(x) - \int \frac{x}{x^2 + 1} \,dx + C$
Evaluating the integral term gives the final result.
$\int \tan^{-1}(x) \,dx = x\tan^{-1}(x) - \frac{1}{2} \ln\left(\left|x^2 + 1\right|\right) + C$
Example: $\int \ln(x) \,dx$
Set up a three column table containing alternating + and − signs, the derivatives of the logarithmic function, and the antiderivatives of unity until the product of a row can be integrated without using integration by parts.
| Sign | D | I |
|---|---|---|
| $+$ | $\ln(x)$ | $1$ |
| $-$ | $\frac{1}{x}$ | $x$ |
The product of the diagonals of the table is equal to $uv$, while the product of the rows of the table is equal to $v \, du$.
$\int \ln(x) \,dx = x\ln(x) - \int dx + C$
Evaluating the integral term gives the final result.
$\int \ln(x) \,dx = x\ln(x) - x + C$
Example: $\int e^{x}\cos(x) \,dx$
Set up a three column table containing alternating + and − signs, the derivatives of the trigonometric function, and the antiderivatives of the exponential function until a row repeats itself.
| Sign | D | I |
|---|---|---|
| $+$ | $\cos(x)$ | $e^x$ |
| $-$ | $-\sin(x)$ | $e^x$ |
| $+$ | $-\cos(x)$ | $e^x$ |
The product of the diagonals of the table is equal to $uv$, while the product of the rows of the table is equal to $v \, du$.
$\int e^{x}\cos(x) \,dx = e^{x}\cos(x) + e^{x}\sin(x) - \int e^{x}\cos(x) \,dx + C$
Add the integral on the right hand side back to the left hand side.
$2\int e^{x}\cos(x) \,dx = e^{x}\cos(x) + e^{x}\sin(x) + C$
Divide by 2 on both sides to obtain the antiderivative we are after.
$\int e^{x}\cos(x) \,dx = \frac{1}{2}\left(e^{x}\cos(x) + e^{x}\sin(x)\right) + C$