Help with definite integral using U-subsitution

Question

I have the following integral I need to use U-subsitution for:

$\int_{0}^{1} \frac{x^{3}}{(x^{2} + 1)^{\frac{3}{2}}}dx$

I'm struggling with mostly the first step. I know I have to rewrite the integral, but I don't how to go about this. After rewriting the integral I'm pretty sure I can manage with substituting, changing limits etc.

Answer 1

Notice that $$ \frac{d}{dx}(x^2+1) = 2x$$ This suggests that we should split off an $x$ from the numerator so that it cancels (after applying the substitution) $$ \int_0^1\frac{x^3}{(x^2+1)^{\frac{3}{2}}}dx = \int_0^1\frac{(x^2+1-1)x}{(x^2+1)^{\frac{3}{2}}}dx $$ so now we let $u=x^2+1\implies du=2xdx$, which gives us $$ \int_1^2\frac{u-1}{2u^{\frac{3}{2}}}du = \frac{1}{2}\int_1^2 (u^{-\frac{1}{2}} - u^{-\frac{3}{2}})du = \dots $$

Answer 2

HINT:

$$\int\dfrac{x^3}{(x^2+1)^{\frac32}}dx=\int\dfrac{x^2\cdot x}{(x^2+1)^{\frac32}}dx$$

Set $\sqrt{x^2+1}=u\implies x^2+1=u^2$ and $\dfrac x{\sqrt{1+x^2}}dx=du$

Answer 3

That $(x^2+1)$ to a power usually suggests a substitution of either $u=x^2+1$ or $x=\tan t$. Both appear to work relatively well, though for the first, you may want to rewrite your integral as

$$\int_0^1\frac{x^3+x-x}{(1+x^2)^{3/2}}dx=\int_0^1\left[\frac{x}{\sqrt{1+x^2}}-\frac{x}{(1+x^2)^{3/2}}\right]dx$$