### Notes from Toppers

**INDEFINITE INTEGRALS**

**1. Properties of Indefinite Integrals**

**Sum, difference, and constant multiples rule:**

For any two functions (f(x)) and (g(x)), and a constant (k), the following hold: $$\int (f(x) + g(x)) \ dx = \int f(x) \ dx + \int g(x) \ dx$$ $$\int (f(x) - g(x)) \ dx = \int f(x) \ dx - \int g(x) \ dx$$ $$\int k f(x) \ dx = k \int f(x) \ dx$$

**Integration of even and odd functions:**

For an even function (f(x)), (\int f(x) \ dx) is also an even function. For an odd function (f(x)), (\int f(x) \ dx) is an odd function.

**2. Integration by Substitution**

Let (u = g(x)) be a differentiable function. Then, $$\int f(g(x)) g’(x) \ dx = \int f(u)\ du$$

*Standard functions:*

- (\int sin(x) \ dx = -\cos(x) + C)
- (\int cos(x) \ dx = \sin(x) + C)
- (\int tan(x) \ dx = \ln|sec(x)| + C)
- (\int sec(x) \ dx = \ln|sec(x) + \tan(x)| + C)
- (\int cosec(x) \ dx = -\ln|cosec(x) - \cot(x)| + C)
- (\int cot(x) \ dx = \ln|sin(x)| + C)

**3. Integration by Parts**

For two functions (u(x)) and (v(x)), the formula for integration by parts is: $$\int u(x) v’(x) \ dx = u(x) v(x) - \int v(x) u’(x) \ dx$$ The choice of (u(x)) and (v(x)) is crucial for successful integration.

**4. Integration of Rational Functions**

*Proper rational functions:*

A rational function is proper if the degree of the numerator is less than the degree of the denominator.

*Improper rational functions:*

A rational function is improper if the degree of the numerator is greater than or equal to the degree of the denominator.

*Integration by partial fractions:*

For an improper rational function, the method of partial fractions involves expressing the function as a sum of simpler fractions, each of which can be easily integrated.

**5. Integration of Irrational Functions**

*Trigonometric substitutions:*

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

*Logarithmic substitutions:*

(\int \sqrt[n]{x} \ dx = \frac{2}{n+1} x^{\frac{n+1}{2}} + C) (\int \frac{1}{x \sqrt{x}} \ dx = 2 \sqrt{x} + C)

*Hyperbolic substitutions:*

For (\int \sqrt{x^2 + 1} \ dx), let (x = \sinh(t)). Then, (dx = \cosh(t) \ dt) and (x^2 + 1 = \cosh^2(t)), so $$\int \sqrt{x^2 + 1} \ dx = \int \cosh(t) \cosh(t) \ dt = \int \frac{1}{2} (\cosh(2t) + 1) \ dt$$ $$\frac{1}{2} \sinh(2t) + \frac{1}{2} t + C$$

Substituting back (t = \ln(x + \sqrt{x^2 + 1})), we get the final result: $$\frac{1}{2} (x + \sqrt{x^2 + 1}) \sqrt{x^2 + 1} + \frac{1}{2} \ln(x + \sqrt{x^2 + 1}) + C$$

**6. Integration of Trigonometric Functions**

*Integration of basic trigonometric functions:*

(\int sin(x) \ dx = -cos(x) + C) (\int cos(x) dx = sin(x) + C) (\int tan(x) dx = \ln|sec(x)| + C) (\int sec(x) dx = \ln|sec(x) + tan(x)| + C) (\int cosec(x) dx = -\ln|cosec(x) - cot(x)| + C) (\int cot(x) dx = \ln|sin(x)| + C)

*Integration of products and powers of trigonometric functions:*

Using trigonometric identities and the above formulas, integrals involving products and powers of trigonometric functions can be simplified and integrated.

**7. Integration of Exponential and Logarithmic Functions**

- (\int e^x \ dx = e^x + C)
- (\int ln(x) \ dx = x \ln(x) - x + C)
- (\int e^{ax + b} \ dx = \frac{1}{a} e^{ax + b} + C)

**8. Improper Integrals**

**Convergence and divergence:**

An improper integral is said to converge if it has a finite value, and it is said to diverge if it does not have a finite value.

**Comparison test and limit comparison test:**

To determine the convergence or divergence of an improper integral, these two tests compare the given integral with another integral whose convergence or divergence is known.

** References**:

- NCERT Textbook for Class 11: Chapter 13 - Integrals
- NCERT Textbook for Class 12: Chapter 7 - Integrals