## What is an antiderivative?

An antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) = f(x)$. In other words, the derivative of $F(x)$ is $f(x)$.

## Why are antiderivatives important?

Antiderivatives are important for a number of reasons. For example, they can be used to:

• Find the area under a curve
• Determine the velocity and displacement of an object given its acceleration
• Solve differential equations

## How to find the antiderivative of a function

There are a number of different ways to find the antiderivative of a function. Some of the most common methods include:

• The power rule: The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$, where $n\neq -1$.
• The sum rule: The antiderivative of $f(x)+g(x)$ is the antiderivative of $f(x)$ plus the antiderivative of $g(x)$.
• The difference rule: The antiderivative of $f(x)-g(x)$ is the antiderivative of $f(x)$ minus the antiderivative of $g(x)$.
• The constant multiple rule: The antiderivative of $kf(x)$ is $k$ times the antiderivative of $f(x)$, where $k$ is any constant.
• Integration by substitution: This technique is used to integrate functions that can be transformed into another function whose integral is already known.
• Integration by parts: This technique is used to integrate products of two functions.
• Integration by trigonometric identities: This technique is used to integrate trigonometric functions.
• Integration by rational functions: This technique is used to integrate rational functions.

## Common integration techniques

Here are some examples of common integration techniques:

• Integration by substitution: To integrate by substitution, we first need to find a function $u(x)$ such that $u'(x) = f(x)$. We then substitute $u(x)$ into the integral and integrate the resulting expression.
• Integration by parts: To integrate by parts, we first need to choose two functions $u(x)$ and $v(x)$. We then multiply $u(x)$ by $v'(x)$ and subtract the product of $u'(x)$ and $v(x)$. The resulting expression is then integrated.
• Integration by trigonometric identities: To integrate by trigonometric identities, we first need to rewrite the trigonometric function in the integral using trigonometric identities. We then integrate the resulting expression.
• Integration by rational functions: To integrate by rational functions, we first need to factor the rational function in the integral. We then integrate the resulting expression using partial fractions.

## Examples of finding antiderivatives

Here are some examples of how to find the antiderivatives of common functions:

• Finding the antiderivative of $x^2$:
∫ x^2 \, dx = x^3 / 3 + C

• Finding the antiderivative of $e^x$:
∫ e^x \, dx = e^x + C

• Finding the antiderivative of $\sin(x)$:
∫ sin(x) \, dx = -cos(x) + C

• Finding the antiderivative of $\frac{1}{x}$:
∫ 1 / x \, dx = ln(x) + C


Conclusion

Antiderivatives are an important part of calculus and have a wide range of applications. In this article, we have covered the basics of antiderivatives, common integration techniques, and examples of finding antiderivatives.

## FAQs

### Q.What is the difference between an antiderivative and a definite integral?

An antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) = f(x)$. A definite integral of a function $f(x)$ over the interval $[a,b]$ is the difference between the antiderivatives of $f(x)$ evaluated at $b$ and $a$.

### Q.How do I find the antiderivative of a function that is not listed in the common integration techniques?

If you are unable to find the antiderivative of a function using the common integration techniques, you can try using a computer algebra system (CAS) such as Wolfram Alpha or Maple. CASs can solve a wide range of mathematical problems, including finding antiderivatives.