# How to solve the integral $\int \frac{4+12x}{({1-2x-3x^2})^\frac{2}{3}}\,dx$

$$\int \frac{4+12x}{({1-2x-3x^2})^\frac{1}{3}}\,dx$$

So first I tried setting $u = 1-2x-3x^2, du = -2-6x\,dx$

Which gives

$$-2\int u^\frac{-1}{3}du = \frac{-6 u^\frac{2}{3}}{2} =-3u^\frac{2}{3}$$

And ultimately:

$$-3(1-2x-3x^2)^\frac{2}{3}$$

For some odd reason when I take the derivative I'm getting something completely different, what's wrong with my substitution?

from $$du=(-2+6x)dx$$ you get $$dx=\frac{du}{-6x-2}$$ but you must express the $x$ in the denominator by $u$ can you solve your integral now? the solution is given by a hypergeometric function

• Indeed! I thought I could get away without solving for dx. – Trey Jan 26 '18 at 15:22
• ok should i give the solution to you here? – Dr. Sonnhard Graubner Jan 26 '18 at 15:23
• That won't be necessary, I but if you want to... – Trey Jan 26 '18 at 15:26
• it is just $$-3 \left(-3 x^2-2 x+1\right)^{2/3}+C$$ – Dr. Sonnhard Graubner Jan 26 '18 at 15:30
• Oh, then my answer was correct. I might have taken the derivative the wrong way – Trey Jan 26 '18 at 15:34

\begin{align} & \frac d {dx} \left( -3(1-2x-3x^2)^{2/3} \right) \\[10pt] = {} & -3\cdot \frac 2 3(1-2x-3x^2)^{(2/3)-1}(-2-6x) & & \text{by the chain rule} \\[10pt] = {} & -2(1-2x-3x^2)^{-1/3}(-2-6x) \\[10pt] = {} & (1-2x-3x^2)^{-1/3} (4+12x) \end{align}

Let's take the derivative:

$$g'(x)=(-3(1-2x-3x^2)^\frac{2}{3})'=-3 \frac 2 3 (1-2x-3x^2)^{\frac {-1}3}(-6x-2)$$ $$g'(x)=2 (1-2x-3x^2)^{\frac {-1}3}(6x+2)=\frac {12x+4}{(1-2x-3x^2)^{\frac {1}3}}$$

Let $u=1-2x-3x^2$, then $\mathrm{d}u=-\left(6x+2\right)\mathrm{d}x$ \begin{align} \int\frac{4+12x}{(1-2x-3x^2)^{1/3}}\,\mathrm{d}x &=\int\frac{-2\,\mathrm{d}u}{u^{1/3}}\\ &=-3u^{2/3}+C\\[3pt] &=-3\left(1-2x-3x^2\right)^{2/3}+C \end{align}

Looking at the answer in the question, this is pretty much exactly what was done. So the question is about what happened when the derivative was taken. \begin{align} -3\frac{\mathrm{d}}{\mathrm{d}x}\left(1-2x-3x^2\right)^{2/3} &=-2\left(1-2x-3x^2\right)^{-1/3}\frac{\mathrm{d}}{\mathrm{d}x}\left(1-2x-3x^2\right)\\ &=\frac{4+12x}{\left(1-2x-3x^2\right)^{1/3}} \end{align} It's hard to tell what went wrong when you computed the derivative, since that wasn't shown.