Question

Evaluate the integral

$\displaystyle\int {x}^{4} \ln \: x \: dx$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm :\longmapsto\:\displaystyle\int \: {x}^{4} \: ln \: x \: dx$

We know, Integration by Parts

$\displaystyle\int \: uvdx =u \displaystyle\int \: vdx - \displaystyle\int \: \bigg[\dfrac{d}{dx} u\displaystyle\int \: vdx\bigg]dx$

So, using this formula taking

$\rm :\longmapsto\:u = {x}^{4}$

$\rm :\longmapsto\:v = ln \: x$

So, on substituting the values, we get

$\rm \:  = ln \: x\displaystyle\int \: {x}^{4} dx - \displaystyle\int \: \bigg[\dfrac{d}{dx} ln \: x\displaystyle\int \: {x}^{4} dx\bigg]dx$

$\rm \:  =  \: ln \: x \: \dfrac{ {x}^{4 + 1} }{4 + 1} - \displaystyle\int \: \dfrac{1}{x} \times \dfrac{ {x}^{4 + 1} }{4 + 1} dx$

$\rm \:  =  \: \dfrac{ {x}^{5} ln \: x}{5} - \displaystyle\int \: \dfrac{1}{x} \times \dfrac{ {x}^{5} }{5} dx$

$\rm \:  =  \: \dfrac{ {x}^{5} ln \: x}{5} - \displaystyle\int \: \dfrac{ {x}^{4} }{5} dx$

$\rm \:  =  \: \dfrac{ {x}^{5} ln \: x}{5} -\dfrac{1}{5} \displaystyle\int \: {x}^{4} dx$

$\rm \:  =  \: \dfrac{ {x}^{5} ln \: x}{5} -\dfrac{1}{5} \times \dfrac{ {x}^{4 + 1} }{4 + 1} + c$

$\rm \:  =  \: \dfrac{ {x}^{5} ln \: x}{5} -\dfrac{1}{5} \times \dfrac{ {x}^{5} }{5} + c$

$\rm \:  =  \: \dfrac{ {x}^{5} ln \: x}{5} - \dfrac{ {x}^{5} }{25} + c$

Therefore,

$\red{\rm \implies\: \boxed{\displaystyle\int \: {x}^{4} \: ln \: x \: dx  =  \: \dfrac{ {x}^{5} ln \: x}{5} - \dfrac{ {x}^{5} }{25} + c}}$

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Basic Concept Used

Integration by Parts

Rule :-

$\displaystyle\int \: uvdx =u \displaystyle\int \: vdx - \displaystyle\int \: \bigg[\dfrac{d}{dx} u\displaystyle\int \: vdx\bigg]dx$

where,

• u is the function u(x)

• v is the function v(x)

• u' is the derivative of the function u(x)

For integration by parts , the ILATE rule is used to choose u and v.

where,

• I - Inverse trigonometric functions

• L -Logarithmic functions

• A - Arithmetic and Algebraic functions

• T - Trigonometric functions

• E- Exponential functions

The alphabet which comes first is choosen as u and other as v.

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More to know :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$