How do you integrate $\int {ln x}^{3} d x$ $int ln x^3 dx$ using integration by parts?

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How do you integrate $\int {ln x}^{3} d x$ $int ln x^3 dx$ using integration by parts?

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Answer 1 · 2016-04-19T22:10:02

$\int ln (x^{3}) d x = 3 x ln x - 3 x + C$ $intln(x^3)dx=3xlnx-3x+C$

Explanation:

First, simplify the function using the rule $ln (a^{b}) = b \cdot ln a$ $ln(a^b)=b*lna$ ; this gives the simplified integral

$\int ln (x^{3}) d x = \int 3 ln x d x = 3 \int ln x d x$ $intln(x^3)dx=int3lnxdx=3intlnxdx$

The question then becomes, how do we integrate $\int ln x d x$ $intlnxdx$ by parts?

Integration by parts takes the form

$\int u d v = u v - \int v d u$ $intudv=uv-intvdu$

So, we have to determine what to let $u$ $u$ and $d v$ $dv$ equal within $\int ln x d x$ $intlnxdx$ . Our best bet is to let $u = ln x$ $u=lnx$ and simply have $d v = 1 d x$ $dv=1dx$ , since there's not much else we can do.

Find the values of $d u$ $du$ and $v$ $v$ , by differentiating and integrating respectively.

$ul(u=lnx)" "=>" "(du)/dx=1/x" "=>" "ul(du=1/xdx)$

$ul(dv=1dx)" "=>" "intdv=intdx" "=>" "ul(v=x$

Plugging these into the integration by parts formula, not forgetting the $3$ tagging along multiplicatively, we have:

$3intlnxdx=3(xlnx-intx(1/x)dx)$

$=3xlnx-3intdx$

$=3xlnx-3x+C$

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How do you integrate $\int {ln x}^{3} d x$ $int ln x^3 dx$ using integration by parts?

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