# Finding antiderivatives

• Apr 7th 2010, 10:08 AM
Schdero
Finding antiderivatives
The antiderivative of the following functions are sought:
1)(x)^1/2. The solution is, in my booklet, 1/2(x)^1/2
But if you use the quotient rule ( f(x)/g(x)' => f'(x)g(x)-f(x)g'(x)/ (g(x))^2)
you get 1/-4x(x)^1/2. Where's the problem? I know, that if you consider 1/(x)^1/2 as x^-1/2 it works out, i just dont see the mistake with applying the quotient rule

2) 2(x^3)^1/2 Here I'm clueless, how do you find the antiderivative at this example?

3) 3sin(x). to me the solution is 3(-cos(x)). In my booklet the solution is 3(cos(x)), but isn't the derivative of cos -> -sinus?

Thanks for you help

Schdero
• Apr 7th 2010, 10:15 AM
harish21
Quote:

Originally Posted by Schdero
The antiderivative of the following functions are sought:
1)(x)^1/2. The solution is, in my booklet, 1/2(x)^1/2
But if you use the quotient rule ( f(x)/g(x)' => f'(x)g(x)-f(x)g'(x)/ (g(x))^2)
you get 1/-4x(x)^1/2. Where's the problem? I know, that if you consider 1/(x)^1/2 as x^-1/2 it works out, i just dont see the mistake with applying the quotient rule

2) 2(x^3)^1/2 Here I'm clueless, how do you find the antiderivative at this example?

3) 3sin(x). to me the solution is 3(-cos(x)). In my booklet the solution is 3(cos(x)), but isn't the derivative of cos -> -sinus?

Thanks for you help

Schdero

Antiderivative means integrating the given function. So, the antiderivative of $x^{\frac{1}{2}}$ would be

$\int x^{\frac{1}{2}} dx = \frac{2}{3} \times x^{\frac{3}{2}} + C$

If you take the derivative of $\frac{2}{3} \times x^{\frac{3}{2}} + C$ with respect to x, you will get what you integrated, that is, $x^{\frac{1}{2}}$

Likewise, the derivative of $3sinx = 3 cos(x)$, and the antiderivative of $3 sinx = -3cosx+C$

Your answer to find the antiderivative of 3sinx is correct! If what you have asked for (that is finding the antiderivative) is correct, may be your booklet has serious flaws!
• Apr 7th 2010, 10:27 AM
Schdero
Quote:

Originally Posted by harish21
If you take the derivative of $\frac{2}{3} \times x^{\frac{3}{2}} + C$ with respect to x, you will get what you integrated.

Sorry I dount understand anything of this.
1)Is $\frac{1}{2(x^{0.5})}$ as antidervative of $x^{0.5}$ henceforth wrong?
2)why does the quotient rule appereantly not apply here or made i a mistake?