IIT JEE Integration by Parts

Indefinite Integral:

An Integral form ∫f(z)dz without upper and lower limits is also called an anti-derivative. The theorem of calculus allows definite integrals to be computed in terms of indefinite integrals. As per this theorem, it is stated that if F is the indefinite integral for a complex function f(z), then ∫abf(z)dz=F(b)-F(a)

An Overview

This article contains an explanation of integration by parts and how to use the ILATE (inverse, algebraic, logarithm, trigonometry, exponent) rule in integration. The frequently asked questions at the end of this article can help you if you incur any doubts while reading the same.

Integration by parts is a procedure used to integrate and it is quite beneficial when two functions are multiplied together, but is also helpful in other ways. There are plenty of examples, but first, it is important to clear the rule:

\[\int uv{}'dx = uv - \int u{}'v dx\]

u is the function u(x)

v is the function v(x)

u' is the derivative of the function u(x)

If u and v are any two differentiable functions in a particular single variable x; then using the product rule of differentiation, we will get:

\[\frac{d}{dx}\left ( u\left ( x \right )v\left ( x \right ) \right )=v\left ( x \right )\frac{d}{dx}\left ( u\left ( x \right ) \right )+u\left ( x \right )\frac{d}{dx}\left ( v\left ( x \right ) \right )\]

Integrating both sides with respect to x

\[\int \frac{d}{dx}\left ( u\left ( x \right ) v\left ( x \right )\right )dx=\int u{}'\left ( x \right )v\left ( x \right )dx+\int u\left ( x \right )v{}'\left ( x \right )dx\]

Then applying the definition of indefinite integers

\[u\left ( x \right )v\left ( x \right )=\int u{}'\left ( x \right )v\left ( x \right )dx+\int u\left ( x \right )v{}'\left ( x \right )dx\]

\[\int u\left ( x \right )v{}'\left ( x \right )dx =u\left ( x \right )v\left ( x \right )- \int u{}'\left ( x \right )v\left ( x \right )dx\]

Gives the formula for integration by parts. Since du and dv are differentials of a function of one variable x. 

\[du=u{}'\left ( x \right )dx dv=v{}'\left ( x \right )dx\]

\[\int u\left ( x \right )dv = u\left ( x \right )v\left ( x \right )=\int v\left ( x \right )du\]

Refer to the official website of Vedantu or download the app for an elaborate and comprehensive explanation.

ILATE Rule

The term ILATE means:

I stands for Inverse trigonometric functions

L means Logarithmic function

A means Algebraic functions

T means Trigonometric functions 

and E means Exponential functions

The preference order of this rule will be dependent on some functions like Inverse, Algebraic, Logarithm, Trigonometric, Exponent, according to the ILATE rule.

Integration by Parts of UV Formula

As mentioned above, integration by parts uv formula is:

\[\int udv=uv-\int vdu\]

Where,

u = Function of u(x)

v = Function of v(x)

dv = Derivative of v(x)

du = Derivative of u(x)

It is also possible to get the formula of integration by parts with limits. Thus, the formula for the same would be:

\[\int_{a}^{b}u dv=uv\int_{a}^{b}-\int_{a}^{b}v du\]

Here,

a = Lower limit

b = Upper limit