Question

How do you integrate two variables?

Answer 1

Integrating two variables:

Integrate two variables also known as integration by parts. We integrate two variables by integration by parts method.

Let $u$ and $v$ be two variables such that $u$is the first function and $v$ is the second function, when integrated with respect to $x$ then

$\mathbf{\int }\mathit{u}\mathbf{\cdot }\mathit{v}\mathit{d}\mathit{x}\mathbf{=}\mathit{u}\mathbf{\int }\mathit{v}\mathit{d}\mathit{x}\mathbf{-}\mathbf{\int }\mathbf{(}\frac{\mathbf{d}\mathbf{u}}{\mathbf{d}\mathbf{x}}\mathbf{\int }\mathit{v}\mathit{d}\mathit{x}\mathbf{)}\mathit{d}\mathit{x}\mathbf{+}\mathit{c}$ where $c$ is a constant.

While using the integration by parts formula, we choose the first function in the following order from top to bottom. This can be remembered using the rule ILATE.

• Inverse trigonometric (I)
• Logarithmic (L)
• Algebraic (A)
• Trigonometric (T)
• Exponential (E)

For eg. Let $u=x$ and $v=e^x$ then according to ILATE rule $u$ is the first function and $v$ is the second function.

$\begin{array}{rl}\int x{e}^{x}dx& =x\int e^{x}dx-\int (\frac{dx}{dx}\int e^x)dx\\ & =x{e}^x-\int e^{x}dx\\ & =x{e}^x-e^x+c\end{array}$

where $c$ is a constant.

Hence, $\begin{array}{rl}\int x{e}^{x}dx& =x{e}^x-e^x\end{array}+c$ where $c$ is a constant.