Introduction:

Integrals form an essential part of Class 12 Mathematics, building the foundation for higher studies in calculus and applications in various fields like physics, engineering, and economics. Among integrals, particular functions have unique properties that simplify calculations and provide deeper insights into their behavior. This document serves as a comprehensive guide, explaining the integration of some specific functions, including logarithmic, exponential, trigonometric, and inverse trigonometric functions, with examples and step-by-step solutions.



Understanding the Basics of Integrals:

Integrals, the reverse process of differentiation, help calculate areas under curves, total quantities, and more. In particular, definite integrals yield a numerical value, while indefinite integrals provide a family of functions with a constant of integration (C).

Integration of Logarithmic Functions:

Logarithmic functions, often encountered in growth and decay models, have specific integration rules:

int ln(x), dx = x ln(x) - x + C

Example:

Evaluate int ln(x), dx.

Solution: Using the formula:

ln(x), dx = x ln(x) - x + C

Integration of Exponential Functions:

Exponential functions like ( e^x ) and ( a^x ) are straightforward to integrate:

1. (int e^x , dx = e^x + C )

2. (int a^x, dx = frac{a^x}{ln(a)} + C, { for } a > 0, a neq 1.)

Example:

Evaluate ( int 5^x, dx).

Solution: Using the second formula:

int 5^x, dx = frac{5^x}{ln(5)} + C

Integration of Trigonometric Functions:

Trigonometric integrals frequently appear in physics and engineering. Important formulas include:

1. int sin(x), dx = -cos(x) + C

2. int cos(x) , dx = sin(x) + C

3. int sec^2(x), dx = tan(x) + C

Example:

Evaluate int cos(2x), dx.

Solution: Apply substitution with u = 2x, du = 2dx:

int cos(2x), dx = frac{1}{2} sin(2x) + C

Integration of Inverse Trigonometric Functions:

Inverse trigonometric functions are integrated using specific formulas:

1. int frac{1}{sqrt{1-x^2}}, dx = arcsin(x) + C

2. int frac{1}{1+x^2} , dx = arctan(x) + C

Example:

Evaluate int frac{1}{sqrt{1-x^2}}, dx).

Solution: Using the first formula:

int frac{1}{sqrt{1-x^2}}, dx = arcsin(x) + C

Special Integrals and Tricks:

Some special forms simplify integration:

1. int e^{ax} sin(bx) \, dx \) and \( \int e^{ax} \cos(bx) \, dx \) use by-parts techniques.

2. \( \int \frac{1}{x \ln(x)} \, dx = \ln(\ln(x)) + C \)

Example:

Evaluate \( \int e^{3x} \cos(2x) \, dx \).

Solution: Use integration by parts twice and simplify.

Conclusion:

Understanding the integration of specific functions is crucial for excelling in Class 12 Mathematics and beyond. By mastering these formulas and practicing their applications, you build a strong foundation in calculus. Regular practice and application of these concepts will help you solve complex problems with confidence.