Shortcut Methods

Indefinite Integrals & Differential Formula in JEE Exam

Indefinite Integrals

JEE Mains Level:

$\int (x^{n}) d x = \frac{x^{n + 1}}{n + 1} + C$
$\int (e^{x}) d x = e^{x} + C$
$\int (\frac{1}{x}) d x = l n | x | + C$
$\int (s i n (x)) d x = - c o s (x) + C$
$\int (c o s (x)) d x = s i n (x) + C$
$\int (t a n (x)) d x = l n | s e c (x) | + C$
$\int (c o t (x)) d x = l n | s i n (x) | + C$
$\int (s e c (x)) d x = l n | s e c (x) + t a n (x) | + C$
$\int (c s c (x)) d x = l n | c s c (x) + c o t (x) | + C$

JEE Advanced Level:

$\int (\sqrt{a^{2} - x^{2}}) d x = \frac{1}{2} x \sqrt{a^{2} - x^{2}} + \frac{1}{2} a^{2} a r c s i n (\frac{x}{a}) + C$
$\int (\frac{1}{\sqrt{a^{2} - x^{2}}}) d x = a r c s i n (\frac{x}{a}) + C$
$\int (\sqrt{x^{2} + a^{2}}) d x = \frac{1}{2} x \sqrt{x^{2} + a^{2}} + \frac{1}{2} a^{2} l n | x + \sqrt{x^{2} + a^{2}} | + C$
$\int (\frac{1}{\sqrt{x^{2} + a^{2}}}) d x = l n | x + \sqrt{x^{2} + a^{2}} | + C$

Differential Formulas for CBSE Exams

1. The derivative of a constant (e.g., 3) is 0.

2. The derivative of x^n (where n is a real number) is nx^(n-1).

3. The derivative of e^x is e^x.

4. The derivative of ln(x) is 1/x.

5. The derivative of sin(x) is cos(x).

6. The derivative of cos(x) is -sin(x).

7. The derivative of tan(x) is sec^2(x).

8. The derivative of cot(x) is -csc^2(x).

9. The derivative of sec(x) is sec(x)tan(x).

10. The derivative of csc(x) is -csc(x)cot(x).

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