Basic Integration Rules

Namaste, JEE aspirants! Welcome to the world of integral calculus. Integration is a fundamental concept, crucial for solving a wide range of problems in JEE Main, from finding areas under curves to solving differential equations. Mastering basic integration rules is your first step towards acing calculus. Let's dive in!

1. Antiderivative: The Reverse Process

Imagine you have a function, say $f(x) = 2x$ . What function, when differentiated, gives you $2x$ ? The answer is $x^2$ (because the derivative of $x^2$ is $2x$ ). We call $x^2$ an antiderivative of $2x$ .

Integration is essentially finding the antiderivative. However, there's a slight twist. The derivative of $x^2 + 1$ is also $2x$ . Similarly, the derivative of $x^2 - 5$ is $2x$ . In fact, the derivative of $x^2 + C$ , where $C$ is any constant, is always $2x$ .

Therefore, the antiderivative is not unique; it's a family of functions differing by a constant. We represent this by adding "+ C" to the antiderivative. This "C" is called the constant of integration. So, the integral of $2x$ is $x^2 + C$ .

2. Power Rule: Integrating Polynomials

The power rule is your go-to tool for integrating polynomial functions. It's a direct consequence of the power rule in differentiation, but used in reverse.

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1

Explanation: To integrate $x^n$ , increase the power by 1 (making it $n+1$ ) and then divide by the new power ( $n+1$ ). Don't forget the "+ C".

Example: Let's integrate $x^3$ . Using the power rule: $\int x^3 \, dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C$

Why $n \neq -1$ ? If $n = -1$ , we have $\int x^{-1} \, dx = \int \frac{1}{x} \, dx$ . The power rule doesn't apply here, as it would lead to division by zero. This case has its own special rule...

3. Integration of 1/x: The Natural Log

The integral of $1/x$ (or $x^{-1}$ ) is the natural logarithm of the absolute value of $x$ .

\int \frac{1}{x} \, dx = \ln|x| + C

Explanation: The derivative of $\ln(x)$ is $1/x$ for $x > 0$ . To handle negative values of $x$ , we use the absolute value. The derivative of $\ln(-x)$ is also $1/x$ for $x < 0$ . Thus, using $\ln|x|$ covers both positive and negative cases.

Example: $\int \frac{1}{x} \, dx = \ln|x| + C$

4. Integrating Exponential Functions

Exponential functions are those where the variable appears in the exponent. Integration rules vary slightly based on the base of the exponential.

a) Integrating $e^x$

\int e^x \, dx = e^x + C

Explanation: The derivative of $e^x$ is $e^x$ itself! Therefore, the integral is also $e^x + C$ .

Example: $\int e^x \, dx = e^x + C$

b) Integrating $a^x$ (where a is a constant)

\int a^x \, dx = \frac{a^x}{\ln a} + C

Explanation: The derivative of $a^x$ is $a^x \ln a$ . Therefore, to get $a^x$ upon differentiation, we need to divide by $\ln a$ during integration.

Example: Let's integrate $2^x$ : $\int 2^x \, dx = \frac{2^x}{\ln 2} + C$

5. Integrating Trigonometric Functions

Integrating trigonometric functions involves recognizing derivative-antiderivative relationships.

\int \sin x \, dx = -\cos x + C

Explanation: The derivative of $-\cos x$ is $\sin x$ .

\int \cos x \, dx = \sin x + C

Explanation: The derivative of $\sin x$ is $\cos x$ .

\int \sec^2 x \, dx = \tan x + C

Explanation: The derivative of $\tan x$ is $\sec^2 x$ .

\int \csc^2 x \, dx = -\cot x + C

Explanation: The derivative of $-\cot x$ is $\csc^2 x$ .

Important Note: Integrals of $\tan x$ , $\cot x$ , $\sec x$ , and $\csc x$ require more advanced techniques (often involving substitution or trigonometric identities) and are usually covered later. For now, focus on these four basic integrals.

Tips for Solving Integration Problems

Simplify first: Before integrating, simplify the expression as much as possible using algebraic identities or trigonometric identities.
Break it down: If the integral is a sum or difference of terms, integrate each term separately. $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$
Check your answer: Differentiate your result. If you get back the original integrand, you've likely integrated correctly.

Common Mistakes to Avoid

Forgetting "+ C": Always include the constant of integration. It represents the entire family of antiderivatives.
Incorrect application of the power rule: Make sure $n \neq -1$ . Also, double-check that you've added 1 to the exponent and divided by the *new* exponent.
Sign errors: Pay close attention to signs, especially when integrating trigonometric functions. Remember that $\int \sin x \, dx = -\cos x + C$ .
Assuming $\int a^x dx = a^x + C$ : Remember to divide by $\ln a$ when integrating $a^x$ .

JEE Specific Tricks

While these basic rules are fundamental, JEE problems often require clever manipulation and application. Here's a common trick:

Trick: Recognizing patterns. JEE questions often involve integrals that *look* complicated but simplify nicely using trigonometric or algebraic identities. Practice recognizing these patterns.

Example: $\int \frac{1}{\sin^2 x + \cos^2 x} \, dx$ might seem tricky, but since $\sin^2 x + \cos^2 x = 1$ , the integral simplifies to $\int 1 \, dx = x + C$ .

Keep practicing, and you'll master these basic integration rules! Good luck with your JEE prep!