Calculus - MAT 140 (UFV)

In this post, we shall be posting some problems solved,

1.  $\int \frac{1}{x\sqrt{x^3-1}}dx$

Note that, $I = \int \frac{x^2}{x^3\sqrt{x^3-1}}dx$

Set $u=x^3-1$.

Hence $du=3x^2 \ dx$.

As a result we have,

$$\begin{eqnarray} I&=&1/3\int\frac{du}{\sqrt{u}(u+1)}\\ &=& 2/3\int \frac{v\ dv}{v(v^2+1)} \mbox { (Substituting } u=v^2 \mbox{we get } du = 2v\ dv)\\ &=&(2/3)\tan^{-1}(\sqrt{x^3-1}) \end{eqnarray}$$

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