What is $\int u^{3} - ln u (\frac{d u}{d x})$ $int u^3-lnu ((du)/dx)$ if $u = 3 x$ $u=3x$ ?

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What is $\int u^{3} - ln u (\frac{d u}{d x})$ $int u^3-lnu ((du)/dx)$ if $u = 3 x$ $u=3x$ ?

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Answer 1 · 2017-09-26T21:20:31

Given: $\int u^{3} - ln | u | (\frac{d u}{d x}); u = 3 x$ $int u^3-ln|u| ((du)/dx); u = 3x$ then $\frac{d u}{d x} = 3$ $(du)/dx = 3$

Moving the multiplication by $\frac{d u}{d x}$ $(du)/dx$ in front of the logarithm:

$\int (u^{3} - 3 ln | u |) d u = \frac{u^{4}}{4} + (3 u) (ln | u | - 1) + C$ $int (u^3-3ln|u|) du = u^4/4+(3u)(ln|u|-1)+ C$

Reversing the substitution for u:

$\int (u^{3} - 3 ln | u |) d u = \frac{{(x^{3})}^{4}}{4} + (3 (x^{3})) (ln ∣ ∣ 3^{3} ∣ ∣ - 1) + C$ $int (u^3-3ln|u|) du = (x^3)^4/4+(3(x^3))(ln|3^3|-1)+ C$

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What is $\int u^{3} - ln u (\frac{d u}{d x})$ $int u^3-lnu ((du)/dx)$ if $u = 3 x$ $u=3x$ ?

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