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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:

\[ \space F’ \space = \space f \]

Expert Answer

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:

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

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