$ \int \frac{1}{2+\sqrt{x+1}+\sqrt{3-x}} \cdot d x $

Question

I am trying to evaluate this antiderivative $$ \int \frac{1}{2+\sqrt{x+1}+\sqrt{3-x}} \cdot d x $$ What i have done:

$$ \begin{split} I &= \int \frac{1}{2+\sqrt{x+1}+\sqrt{3-x}} \cdot d x \\ &= \int \frac{2+\sqrt{x+1}-\sqrt{3-x}}{4+x+1+4 \sqrt{x+1}-3+x} \cdot d x\\ &= \int \frac{2+\sqrt{x+1}-\sqrt{3-x}}{2+2 x+4 \sqrt{x+1}} \cdot d x\\&=\int \frac{1}{1+x+2 \sqrt{x+1}} \cdot d x\\ &\quad +\int \frac{\sqrt{x+1}}{2(1+x)+4 \sqrt{x+1}}\cdot d x \\ &\quad -\int\frac{\sqrt{3-x}}{2(1+x)+4 \sqrt{x+1}} \cdot d x \end{split} $$ From this last line i can evaluate the first two antiderivatives but the last one $$\int\frac{\sqrt{3-x}}{2(1+x)+4 \sqrt{x+1}} \cdot dx$$ seems a bit hard for me two evaluate, i don't find the good substitution. I am opened to any suggestion. Thanks in advance!

Answer 1

Too large for a comment in the other answer.

Let us first try to identify the correct substitution

For real calculus, we need $x+1\ge0\text{ and }3-x\ge0\iff-1\le x\le3$

$$\iff-\dfrac12-\dfrac12\le\dfrac{x-1}2\le\dfrac32-\dfrac12$$

WLOG $\dfrac{x-1}2=\cos2t, 0\le2t\le\pi\implies x=2\cos2t+1$

$\sqrt{x+1}=+(2\cos t),\sqrt{3-x}=+(2\sin t)$ as $0\le t\le\dfrac\pi2$

Finally use $(\sin t+\cos t+1)(\sin t+\cos t-1)=(\sin t+\cos t)^2-1=?$

Answer 2

hint

$$x+1=(x-1)+2$$ $$3-x=2-(x-1)$$

put

$$x-1=2\cos(2t)$$ to get

$$x+1=4\cos^2(t)$$ and $$3-x=4\sin^2(t)$$

the integrale becomes $$\int\frac{-2\sin(2t)dt}{1+\sin(t)+\cos(t)}$$

which can be computed using the substitution $$u=\tan(\frac t2)$$

Answer 3

Solution without trigonometry:

Let $u=\sqrt{x+1}$ and $v=\sqrt{3-x}$, then $u^2+v^2=4$ and $u^2-v^2=2x-2$, so $udu+vdv=0$ and $dx=udu-vdv=2udu=-2vdv$.

\begin{align*}\int\frac{dx}{2+u+v}&=\int\frac{(u+v-2)dx}{(u+v)^2-4}= \int\frac{(u+v-2)dx}{2uv}=\int\left(\frac{dx}{2v}+\frac{dx}{2u}-\frac{dx}{uv}\right)\\&=\int\left(-dv+du-\frac{dx}{uv}\right)=-v+u-\int\frac{du}{v}+\int\frac{dv}{u}\\&=-v+u-\int\frac{(u^2+v^2)du}{4v}+\int\frac{(u^2+v^2)dv}{4u}\\&=-v+u+\int\frac{v(udv-vdu)}{4v}+\int\frac{u(udv-vdu)}{4u}\\&=-v+u+\frac12\int(udv-vdu)=-v+u+2\int\frac{udv-vdu}{u^2+v^2}\\&=-v+u+2\int\frac{u^2d(v/u)}{u^2+v^2}=-v+u+2\arctan(v/u)+C.\end{align*}