Find antiderivative here

**Critter314** · November 15th 2010, 11:31 AM

Is antiderivative of $(6x^2+1)$ is it $2x^3+x+C$ ?
Is antiderivative of $\frac{1}{4z}$ is it $\frac{1}{4}\ln4z+C$ ?

**e^(i*pi)** · November 15th 2010, 11:53 AM

Originally Posted by Critter314

Is antiderivative of $(6x^2+1)$ is it $2x^3+x+C$ ?

Yes (Clap)

Is antiderivative of $\frac{1}{4z}$ is it $\frac{1}{4}\ln4z+C$ ?

Yes (Clap)

**Soroban** · November 15th 2010, 12:10 PM

Hello, Critter314!

$\text{The antiderivative of }\,\dfrac{1}{4z}$

$\text{Is it }\,\frac{1}{4}\ln4z+C\,?$

Well, yes and no . . .

Note that: . $\displaystyle \int\frac{dz}{4z} \;=\;\tfrac{1}{4}\int\frac{dz}{z} \;=\; \tfrac{1}{4}\ln z + C$

Your answer is correct, but it can be simplified.

. . $\frac{1}{4}\ln4z + C \;=\; \frac{1}{4}\bigg[\log 4 + \ln z\bigg] + C$

. . . . . . . . . . . $=\;\underbrace{\tfrac{1}{4}\ln4}_{\text{constant}} + \frac{1}{4}\ln z + C$

. . . . . . . . . . . $=\;\tfrac{1}{4}\ln z + C$

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