# Integration by Parts

### Integration by Parts

If u and v be two functions of x, then the integral of product of these two functions is given by $$\int{uv}.dx=u\int{vdx-\int{\left[ \frac{du}{dx}\int{vdx} \right]}}dx$$.

In applying the above rule care has to be taken in the selection of the first function (u) and the second function (v) depending on which function can be integrated easily. Normally we use the following methods for making this choice:

If both functions are directly integrable then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable. Usually we use the following preference order for the first function.

(Inverse, Logarithmic, Algebraic, Trigonometric, Exponent)

This rule is called as ILATE which states that the inverse function should be assumed as the first function while performing integration. Hence, the functions are to be assumed in the order from left to right depending on the type of functions involved. Various other cases have been discussed in detail in the later sections.

Examples

1. Evaluate∫ xsec²x dx.

Solution: Let I = ∫ xsec²x dx.

= x (tanx) – ∫ tanx dx.

= x tanx – log|secx|+c

2. Evaluate∫logx/x² dx.

Solution: ∫logx/x² dx = (logx)(-1/x) + ∫1/x.1/x dx

=-1/x  logx – 1/x² + c

3. ∫(1ogx)² dx

Solution: ∫(1ogx)² dx = (1ogx)²x – ∫x.2logx.1/x dx

= (1ogx)² x – 2∫logx. dx

= (1ogx)² x – 2(xlogx-∫x.1/x dx)

= (logx)² – 2x.logx + x + c