# What is the antiderivative of the given expression.

– $x^2$

The main objective of this question is to find the anti-derivative of the given expression.

This question uses the concept of anti-derivative. In calculus, if a function $f$ has a derivative, then another differentiable function $F$ with the same derivative is called an antiderivative of $f$. It is represented as:

$\space F’ \space = \space f$

Given that:

$\space = \space x^2$

We have to find the anti-derivate of the given function.

We know that:

$\int x^n \,dx \space = \space \frac{ x^{ n + 1 } }{ n \space + \space 1 } \space + C \space if \space n \space \neq \space – \space 1$

So:

$\space f ( x ) \space = \space x^2$

Let:

$\space F(x) \space = \space \int f(x) ,dx$

Using the above formula results in:

$\space = \space \frac{ x^3 }{3} \space + \space C$

Thus the anti-derivative is:

$\space F ( x ) \space = \space \frac{ x^3 }{3} \space + \space C$

## Numerical Results

The anti-derivative of the given expression is:

$\space F ( x ) \space = \space \frac{ x^3 }{ 3 } \space + \space C$

## Example

Find the anti-derivative of the given expressions.

• $\space x^3$
• $\space x^4$
• $\space x^5$

Given that:

$\space = \space x^3$

We have to find the anti-derivate of the given function.

We know that:

$\int_ x^n \,dx \space = \space \frac{ x^{ n + 1 } }{ n \space + \space 1 } \space + C \space if \space n \space \neq \space – \space 1$

So:

$\space f ( x ) \space = \space x^3$

Let:

$\space F ( x ) \space = \space \int f( x ) ,dx$

Using the above formula results in:

$\space = \space \frac{ x^4 }{ 4 } \space + \space C$

Thus the anti-derivative is:

$\space F ( x ) \space = \space \frac{ x^4 }{ 4 } \space + \space C$

Now for the second expression. Given that:

$\space = \space x^4$

We have to find the anti-derivate of the given function.

We know that:

$\int x^n \,dx \space = \space \frac{ x^{ n + 1 } }{ n \space + \space 1 } \space + C \space if \space n \space \neq \space – \space 1$

So:

$\space f ( x ) \space = \space x^4$

Let:

$\space F( x ) \space = \space \int f ( x ) ,dx$

Using the above formula results in:

$\space = \space \frac{ x^5 }{ 5 } \space + \space C$

Thus the anti-derivative is:

$\space F ( x ) \space = \space \frac{ x^5 }{ 5 } \space + \space C$

Now for the third expression. Given that:

$\space = \space x^5$

We have to find the anti-derivate of the given function.

We know that:

$\int x^n \,dx \space = \space \frac{ x^{ n + 1 } }{ n \space + \space 1 } \space + C \space if \space n \space \neq \space – \space 1$

So:

$\space f ( x ) \space = \space x^5$

Let:

$\space F( x ) \space = \space \int f ( x ) ,dx$

Using the above formula results in:

$\space = \space \frac{ x^6 }{ 6 } \space + \space C$

Thus, the anti-derivative is:

$\space F ( x ) \space = \space \frac{ x^6 }{ 6 } \space + \space C$