(x^3) (x^2 +1)^(1/2)
can anyone tell me what i should substitute here?
$\displaystyle \int{x^3\sqrt{x^2 + 1}\,dx}$.
Try $\displaystyle x = \sinh{t}$ so that $\displaystyle dx = \cosh{t}\,dt$.
The integral becomes
$\displaystyle \int{\sinh^3{t}\sqrt{\sinh^2{t} + 1}\,\cosh{t}\,dt}$
$\displaystyle = \int{\sinh^3{t}\cosh{t}\sqrt{\cosh^2{t}}\,dt}$
$\displaystyle = \int{\sinh^3{t}\cosh^2{t}\,dt}$
$\displaystyle = \int{\sinh^2{t}\cosh^2{t}\sinh{t}\,dt}$
$\displaystyle = \int{(\cosh^2{t} - 1)\cosh^2{t}\sinh{t}\,dt}$
$\displaystyle = \int{(\cosh^4{t} - \cosh^2{t})\sinh{t}\,dt}$.
Now make the substitution $\displaystyle u = \cosh{t}$ so that $\displaystyle du = \sinh{t}\,dt$.
The integral becomes
$\displaystyle \int{u^4 - u^2\,du}$
$\displaystyle = \frac{u^5}{5} - \frac{u^3}{3} + C$
$\displaystyle = \frac{\cosh^5{t}}{5} - \frac{\cosh^3{t}}{3} + C$
$\displaystyle = \frac{(\cosh^2{t})^{\frac{5}{2}}}{5} - \frac{(\cosh^2{t})^{\frac{3}{2}}}{3} + C$
$\displaystyle = \frac{(\sinh^2{t} + 1)^{\frac{5}{2}}}{5} - \frac{(\sinh^2{t} + 1)^{\frac{3}{2}}}{3} + C$
$\displaystyle = \frac{(x^2 + 1)^{\frac{5}{2}}}{5} - \frac{(x^2 + 1)^{\frac{3}{2}}}{3} + C$
$\displaystyle = \frac{\sqrt{(x^2 + 1)^5}}{5} - \frac{\sqrt{(x^2 + 1)^3}}{3} + C$.
Another way to do it:
$\displaystyle \int x^3\sqrt{x^2+ 1} dx= \int (x^2)\sqrt{x^2+ 1} (xdx)$
Let $\displaystyle u= x^2+ 1$ so du= 2x dx and xdx= (1/2)du.
Also, x^2= u- 1.
Now the integral is $\displaystyle \frac{1}{2}\int (u- 1)u^{1/2}du= \frac{1}{2}\int u^{3/2}- u^{1/2} du$
$\displaystyle = \frac{1}{2}\left(\frac{2}{5}u^{5/2}- \frac{2}{3}u^{3/2}\right)+ C$
$\displaystyle = \frac{1}{5}(x^2+ 1)^{5/2}- \frac{1}{3}(x^2+ 1)^{3/2}+ C$
That is, of course, the same as Prove It's answer.
And another...
... where
... is the product rule (straight lines differentiating downwards), so that...
is integration by parts, but (I hope) a lazy version where choosing legs crossed or un-crossed does for assigning u du and v dv.
Spoiler:
_________________________________________
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!