__Integration by Parts __

Integration by parts is a unique fashion of integrating two functions when they’re multiplied together or for evaluating integrals whose integrand is the product of two functions to make complicated integrations easy to break. This system is also nominated as partial integration.

__Formula : Integration by Parts__

The following equation gives the formula for integrating by parts;

$$ \int f(x) g(x) d x=f(x) \int g(x) d x-\int\left[f^{\prime}(x) \int g(x) d x\right] d x $$

__ILATE Rule__

In integration by parts, we’ve learnt when the product of two functions is given to us further, we apply the needed formula. The integral of the two functions is taken by considering the left term as the first function and the other term as the alternate function. This system is called the Ilate rule. Suppose we’ve to integrate $$ x \cos x $$; in this case, we consider x as the first function and cos x as the alternate function. So mainly, the first function is chosen so that the derivative of the function could be fluently integrated. Generally, the preference order of this rule is grounded on some functions similar as Inverse, Algebraic, Logarithm, Trigonometric, and Exponent.

To summarise the way of applying ILATE rule we can follow the below-given steps,

- Identify the type of each function as Inverse trigonometric( I), Logarithmic( L), Algebraic( I), Trigonometric( T), or Exponential( E).
- Now precisely choose which of the given functions comes first in the order of ILATE( or) LIATE and let it be as the first function.
- Elect the remaining function as the alternate function.
- Then apply the integration by parts formula.

For example, Question : Integrate $$ \int x \ln x d x $$

Solution:

Now applying the above-given steps,

- x is the algebraic function and ln x is the logarithmic function.
- Among Algebraic (A) and Logarithmic (L) functions, we know that the Logarithmic (L) function comes first in ILATE rule. So the first function is ln x as per ILATE rule.
- The remaining function, x, is elected as the second function.

By applying the integration by parts formula, we get the following:

$$\begin{aligned}& \int x \ln x d x=(\ln x) \int x d x-\int\left[d / d x(\ln x) \int x d x\right] d x \\& =(\ln x)\left(x^2 / 2\right)-\int(1 / x)\left(x^2 / 2\right) d x \\& =\left(x^2 \ln x\right) / 2-(1 / 2) \int x d x \\& =\left(x^2 \ln x\right) / 2-(1 / 2)\left(x^2 / 2\right) d x \\& =\left(x^2 \ln x\right) / 2-\left(x^2 / 4\right)+C\end{aligned}$$

__Important points regarding Integration by parts:__

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- Integration by parts isn’t applicable for functions similar to $$ \int \sqrt{x} \cos x d x $$
- We don’t add any constant while finding the integral of the alternate function.
- Generally, if any function is a power of x or a polynomial in x, we take it as the first function. Even so, in cases where another function is an inverse trigonometric or logarithmic function, we take them as the first function.
- Apart from the Integration by parts method,we use two other methods which are used to perform the integration. They are:

- Integration by Substitution
- Integration using Partial Fractions