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Question

How do you integrate two variables?


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Solution

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 uis the first function and v is the second function, when integrated with respect to x then

uvdx=uvdx(dudxvdx)dx+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=ex then according to ILATE rule u is the first function and v is the second function.

xexdx=xexdx(dxdxex)dx=xexexdx=xexex+c

where c is a constant.

Hence, xexdx=xexex+c where c is a constant.


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