Antiderivative Formulas

Antiderivative formulas are key tools in Calculus I, helping us find functions whose derivatives we know. These formulas cover a range of functions, from polynomials to trigonometric and exponential functions, each with a constant C representing the family of solutions.

1. ∫ x^n dx = (1/(n+1)) x^(n+1) + C, for n ≠ -1

   • This formula is fundamental for finding the antiderivative of polynomial functions.
   • The exponent n must not be -1 to avoid division by zero.
   • The constant C represents the family of antiderivatives, accounting for any constant term.

2. ∫ 1/x dx = ln|x| + C

   • This formula is used for finding the antiderivative of the reciprocal function.
   • The absolute value is important to handle both positive and negative values of x.
   • The natural logarithm function arises from the properties of exponential growth.

3. ∫ e^x dx = e^x + C

   • The exponential function e^x is unique because it is its own derivative and antiderivative.
   • This formula is crucial in solving differential equations involving exponential growth.
   • The constant C indicates that there are infinitely many antiderivatives differing by a constant.

4. ∫ sin(x) dx = -cos(x) + C

   • The antiderivative of sine results in the negative cosine function.
   • This relationship is essential in trigonometric integration and solving problems involving oscillatory motion.
   • The constant C represents the general solution to the integral.

5. ∫ cos(x) dx = sin(x) + C

   • The antiderivative of cosine is the sine function, highlighting the symmetry in trigonometric functions.
   • This formula is frequently used in physics and engineering applications involving periodic functions.
   • The constant C signifies the family of solutions to the integral.

6. ∫ tan(x) dx = -ln|cos(x)| + C

   • The antiderivative of tangent involves the natural logarithm of the absolute value of cosine.
   • This formula is useful in calculus and trigonometric identities.
   • The negative sign indicates the relationship between tangent and cosine functions.

7. ∫ sec^2(x) dx = tan(x) + C

   • The secant squared function is the derivative of the tangent function.
   • This formula is essential in calculus, particularly in integration involving trigonometric identities.
   • The constant C represents the general solution to the integral.

8. ∫ 1/(1+x^2) dx = arctan(x) + C

   • The antiderivative of the function 1/(1+x^2) leads to the arctangent function.
   • This formula is important in calculus for integrating rational functions.
   • The constant C indicates the family of antiderivatives.

9. ∫ 1/√(1-x^2) dx = arcsin(x) + C

   • The antiderivative of 1/√(1-x^2) results in the arcsine function.
   • This formula is crucial in calculus, especially in problems involving circular functions.
   • The constant C represents the general solution to the integral.

10. ∫ a^x dx = (1/ln(a)) a^x + C, for a > 0 and a ≠ 1

• This formula is used for finding the antiderivative of exponential functions with bases other than e.
• The natural logarithm of the base a is necessary for scaling the function correctly.
• The constant C indicates the family of antiderivatives, similar to other formulas.