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**Reckoner** Don't forget to use the chain rule: $\displaystyle du = 7e^{7x}\,dx$

The power rule for integration is $\displaystyle \int x^n\,dx=\frac{x^{n+1}}{n+1}+C, n\neq-1\text.$ Obviously, this cannot work for an exponent of -1, because then there would be a division by zero. Instead, you have to use the log rule, which Jameson listed above.

Let $\displaystyle u = x^2 + 42\text.$ Then $\displaystyle du = 2x\,dx\Rightarrow dx=\frac{du}{2x}\text.$

Substitution gives

$\displaystyle \int x^3\sqrt{x^2+42}\,dx=\int x^3\sqrt u\left(\frac{du}{2x}\right)$

$\displaystyle =\frac12\int x^2\sqrt u\,du$

But here's the kicker: $\displaystyle u=x^2+42$ means that $\displaystyle x^2 = u-42\text.$ So we can substitute for $\displaystyle x^2$ to get

$\displaystyle \frac12\int(u-42)\sqrt u\,du$

$\displaystyle =\frac12\left[\int u\sqrt u\,du - 42\int\sqrt u\,du\right]$

$\displaystyle =\frac12\left[\int u^{3/2}\,du - 42\int u^{1/2}\,du\right]$