IIT JEE Main Maths -Unit 8- Standard integrals (algebraic, trigonometric, exponential, logarithmic)- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 8- Standard integrals (algebraic, trigonometric, exponential, logarithmic) – Study Notes – New syllabus
IIT JEE Main Maths -Unit 8- Standard integrals (algebraic, trigonometric, exponential, logarithmic) – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Standard Integrals : Algebraic, Trigonometric, Exponential, and Logarithmic Functions
Standard Integrals — Algebraic, Trigonometric, Exponential, and Logarithmic Functions
In integration, some functions have well-known results that are used repeatedly. These are called standard integrals. Knowing these formulas helps solve most indefinite and definite integrals quickly.
Standard Integrals of Algebraic Functions
| Function \( f(x) \) | Integral \( \displaystyle \int f(x)\,dx \) | Conditions |
|---|---|---|
| \( x^n \) | \( \dfrac{x^{n+1}}{n+1} + C \) | \( n \ne -1 \) |
| \( \dfrac{1}{x} \) | \( \ln|x| + C \) | \( x \ne 0 \) |
| \( \dfrac{1}{ax + b} \) | \( \dfrac{1}{a}\ln|ax + b| + C \) | — |
Standard Integrals of Trigonometric Functions
| Function \( f(x) \) | Integral \( \displaystyle \int f(x)\,dx \) |
|---|---|
| \( \sin x \) | \( -\cos x + C \) |
| \( \cos x \) | \( \sin x + C \) |
| \( \sec^2 x \) | \( \tan x + C \) |
| \( \csc^2 x \) | \( -\cot x + C \) |
| \( \sec x \tan x \) | \( \sec x + C \) |
| \( \csc x \cot x \) | \( -\csc x + C \) |
| \( \tan x \) | \( \ln|\sec x| + C \) |
| \( \cot x \) | \( \ln|\sin x| + C \) |
Standard Integrals of Exponential Functions
| Function \( f(x) \) | Integral \( \displaystyle \int f(x)\,dx \) |
|---|---|
| \( e^x \) | \( e^x + C \) |
| \( a^x \) | \( \dfrac{a^x}{\ln a} + C \) |
| \( e^{ax} \) | \( \dfrac{1}{a}e^{ax} + C \) |
Standard Integrals of Logarithmic and Inverse Trigonometric Functions
| Function \( f(x) \) | Integral \( \displaystyle \int f(x)\,dx \) |
|---|---|
| \( \dfrac{1}{1+x^2} \) | \( \tan^{-1}x + C \) |
| \( \dfrac{1}{\sqrt{1 – x^2}} \) | \( \sin^{-1}x + C \) |
| \( \dfrac{-1}{\sqrt{1 – x^2}} \) | \( \cos^{-1}x + C \) |
| \( \dfrac{1}{x\sqrt{x^2 – 1}} \) | \( \sec^{-1}x + C \) |
Example
Find \( \displaystyle \int (4x^3 – 2x + 5)\,dx \).
▶️ Answer / Explanation
Integrate term by term:
\( \displaystyle \int 4x^3\,dx = x^4, \quad \int -2x\,dx = -x^2, \quad \int 5\,dx = 5x \).
Answer: \( x^4 – x^2 + 5x + C \).
Example
Find \( \displaystyle \int e^x(\sin x + \cos x)\,dx \).
▶️ Answer / Explanation
Use standard result: \( \displaystyle \int e^x(\sin x + \cos x)\,dx = e^x \sin x + C \).
(Verified by differentiating \( e^x \sin x \)).
Answer: \( e^x \sin x + C \).
Example
Evaluate \( \displaystyle \int \dfrac{dx}{x^2 + 9} \).
▶️ Answer / Explanation
Rewrite: \( \displaystyle \int \dfrac{dx}{x^2 + 9} = \int \dfrac{dx}{(3)^2 + x^2} \).
Using standard formula: \( \displaystyle \int \dfrac{dx}{a^2 + x^2} = \dfrac{1}{a}\tan^{-1}\left(\dfrac{x}{a}\right) + C \).
Answer: \( \dfrac{1}{3}\tan^{-1}\left(\dfrac{x}{3}\right) + C \).