## Properties of Indefinite Integral for JEE Main and Advanced

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

## FAQs on IIT JEE Integration by Parts

**1. How can I learn integration fast?**

Integration is a very wide topic and it requires command over properties of functions and basic knowledge. A standard book like RD Sharma or any other one to solve a healthy amount of problems that check your analytical skills will be useful. Start with basics and then move forward, there are some predefined forms of integration that are applicable everywhere so you just need to have those forms in mind and Integration is such an interesting topic that if in gulfs so many topics under it such as complex numbers, vectors and many other kinds of stuff. Refer to the official website of Vedantu or download the app for an elaborate and comprehensive explanation.

**2. What are the main types of integration?**

**The main types of integration are:**

**Backward Vertical Integration-**This involves acquiring a business operating earlier in the supply chain.**Conglomerate Integration-**This involves the combination of firms that are involved in unrelated business activities.**Forward Vertical Integration-**This involves acquiring a business further up in the supply chain – e.g. a vehicle manufacturer buys a car parts distributor.**Horizontal Integration-**Here, businesses in the same industry and which operate at the same stage of the production process are combined.

**3. Why do we use integration by parts?**

The integration by parts formula is used to find the integral of the product of two different types of functions such as logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions. The integration by parts formula is used to find the integral of a product. In the product rule of differentiation where we differentiate a product uv, u(x), and v(x) can be chosen in any order.

**4. Is integration by parts the same as U substitution?**

If the integral is simple, you can make a simple tendency behaviour: if you have a composition of functions, u-substitution may be a good idea; if you have products of functions that you know how to integrate, you can try integration by parts. But most difficult integrals have no immediate ideas. Maybe you should use them both. If the integral is simple, you can make a simple tendency behaviour: if you have the composition of functions, u-substitution may be a good idea; if you have products of functions that you know how to integrate, you can try integration by parts.