# Anti derivative

• Apr 23rd 2012, 12:39 AM
tankertert
Anti derivative
Confused guys
Whats the most general anti derivative of

f(x) = 4x + (3/x^2)

And

y(x) = 5x e^(1-x^2) + sin x cos x

Plz thanks
• Apr 23rd 2012, 02:28 AM
Prove It
Re: Anti derivative
Quote:

Originally Posted by tankertert
Confused guys
Whats the most general anti derivative of

f(x) = 4x + (3/x^2)

And

y(x) = 5x e^(1-x^2) + sin x cos x

Plz thanks

For the first one, start by writing \displaystyle \begin{align*} f(x) = 4x + 3x^{-2} \end{align*} and apply the rule \displaystyle \begin{align*} \int{a\,x^n\,dx} = \frac{a\,x^{n+1}}{n+1} \end{align*}, and for the second

\displaystyle \begin{align*} \int{5x\,e^{1 - x^2} + \sin{x}\cos{x}\,dx} &= \int{5x\,e^{1 - x^2}\,dx} + \int{\sin{x}\cos{x}\,dx} \\ &= -\frac{5}{2}\int{-2x\,e^{1 - x^2}\,dx} + \int{\sin{x}\cos{x}\,dx} \end{align*}

Then substitute \displaystyle \begin{align*} u = 1 - x^2 \implies du = -2x\,dx \end{align*} and \displaystyle \begin{align*} v = \sin{x} \implies dv = \cos{x}\,dx \end{align*} :)