Integration by substitution to take out square root

Question

Find $$\int (x+1)\sqrt{x^2+1}\,dx .$$

In order to not bother with the square root I thought of doing this:

$let$ $ x^2+1=(x+t)^2$ $\Rightarrow$ $x=\frac{1-t^2}{2t}$ $\Rightarrow x-t=\frac{1+t^2}{2t}$ $(1)$

$\Rightarrow$ $t=\sqrt{x^2+1}-x$ $(2)$

And

$dx=-\frac{t^2+1}{2t^2}dt $ $(3)$

and also $x+1=-\frac{t^2-2t-1}{2t}$ $(4)$

Then

$(1),(3),(4)\Rightarrow$$\int (x+1)\sqrt{x^2+1}dx=$ $\int\frac{t^2-2t-1}{2t} \frac{1-t^2}{2t} \frac{t^2+1}{2t}dt=$

$\frac{1}{8} \int \frac{-t^6+2t^5+t^4+t^2-2t-1}{t^3}dt=$

$\frac{1}{8} [-\int t^3 dt+2\int t^2 dt+\int t dt +\int \frac{1}{t} dt-2 \int \frac{1}{t^2} dt- \int \frac{1}{t^3}dt]=$

$\frac{1}{8}[-\frac{t^4}{4}+2\frac{t^3}{3}+\frac{t^2}{2}+lnt+\frac{2}{t}-\frac{1}{2t^2}]$

We do the substitution from $(2)$ and we get the result

You think that's correct?

Answer 1

After the rationalizing substitution, I think you made a few errors. It should be $$\int (x+1)\sqrt{x^2+1}\,dx=\int\frac{t^2-2t-1}{2t}\cdot \frac{t^2+1}{2t}\cdot \frac{t^2+1}{2t^2}dt.$$

Alternative approach by using hyperbolic functions.

Let $x=\sinh(t)$ then $\cosh^2(t)-\sinh^2(t)=1$ and $$\begin{align*}\int (x+1)\sqrt{x^2+1}\,dx &=\int (\sinh(t)+1)\sqrt{\sinh^2(t)+1}\,\cosh(t) dt \\ &=\int (\sinh(t)+1)\cosh^2(t) dt\\ &=\frac{\cosh^3(t)}{3}+ \frac{\cosh(t)\sinh(t)}{2}+\frac{t}{2}+c\\ &=\frac{(x^2+1)^{3/2}}{3}+\frac{x(x^2+1)^{1/2}}{2}+\frac{\text{arcsinh}(x)}{2}+c \end{align*}.$$

Answer 2

Before trying the substitution, it is worth to simplify the integrand, by observing that

$$(x^2+1)'=2x$$ so that

$$\int(x+1)\sqrt{x^2+1}\,dx=\frac13(x^2+1)^{3/2}+\int\sqrt{x^2+1}\,dx.$$

For the last integral, substitution with $x=\sinh t$ will allow to get rid of the square root. By parts intégration also works:

$$I:=\int\sqrt{x^2+1}\,dx=x\sqrt{x^2+1}-\int\frac{x^2}{\sqrt{x^2+1}}dx$$

and

$$\int\frac{x^2}{\sqrt{x^2+1}}dx=\int\frac{x^2+1-1}{\sqrt{x^2+1}}dx=I-\text{arshinh } x,$$ from which you draw $I$.

Answer 3

Avoiding hyperbolic functions, I believe you can do a trigonometric substitution: consider a right triangle with angle $\theta$ for which the opposite side is $x$, the adjacent side is $1$, and (thus) the hypotenuse is $\sqrt{x^2 + 1}$.

Then $x = \tan{\theta}$, so $dx = \sec^2{\theta}d\theta$; note that $\sec{\theta} = \sqrt{x^2 + 1}$ and your integral becomes:

$$\int(x+1)\sqrt{x^2 + 1}dx = \int(\tan{\theta} + 1)\sec^3{\theta} d\theta = \int\tan{\theta}\sec^3{\theta} + \sec^3{\theta} d \theta$$

The first part of the integrand is not difficult, since $u = \sec{\theta}$ yields $du = \sec{\theta}\tan{\theta}d\theta$, whereby

$$\int \tan\theta \sec^3\theta d\theta = \int u^2 du$$

and finding the antiderivative of secant cubed is now the only task that remains. If you have seen this before (example link) then you have reduced the problem to something familiar. If not, then you can use the already provided answer(s) for an alternative derivation of this function's antiderivative!