### Shortcut Methods

**Indefinite Integrals & Differential Formula in JEE Exam**

**Indefinite Integrals**

**JEE Mains Level:**

- $$∫(x^n)dx = \frac{x^{n+1}}{n+1} + C$$
- $$∫(e^x)dx = e^x + C$$
- $$∫(\frac{1}{x})dx = ln|x| + C$$
- $$∫(sin(x))dx = -cos(x) + C$$
- $$∫(cos(x))dx = sin(x) + C$$
- $$∫(tan(x))dx = ln|sec(x)| + C$$
- $$∫(cot(x))dx = ln|sin(x)| + C$$
- $$∫(sec(x))dx = ln|sec(x) + tan(x)| + C$$
- $$∫(csc(x))dx = ln|csc(x) + cot(x)| + C$$

**JEE Advanced Level:**

- $$∫(\sqrt{a^2 - x^2})dx = \frac{1}{2}x\sqrt{a^2 - x^2} + \frac{1}{2}a^2arcsin\left(\frac{x}{a}\right) + C$$
- $$∫(\frac{1}{\sqrt{a^2 - x^2}})dx = arcsin\left(\frac{x}{a}\right) + C$$
- $$∫(\sqrt{x^2 + a^2})dx = \frac{1}{2}x\sqrt{x^2 + a^2} + \frac{1}{2}a^2ln|x + \sqrt{x^2 + a^2}| + C$$
- $$∫(\frac{1}{\sqrt{x^2 + a^2}})dx = ln|x + \sqrt{x^2 + a^2}| + C$$

**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).