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