Mathematics

Evaluate:
$$\displaystyle\int{{(\ln x)}^{4}dx}$$


SOLUTION
$$\displaystyle \int (ln \, x)dx$$
Let $$ln\, x = t$$
$$x = e^t$$
$$dx =e^t dt$$

$$\displaystyle  \int t^4e^tdt$$
   
Using integration by parts,
$$\displaystyle \int u.v \ dx=u\int v\ dx$$ $$\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx$$

$$\displaystyle = t^4 \int e^tdt - \int 4t^3e^tdt$$

$$\displaystyle = t^4 e^t - 4 \left[t^3 \int e^t dt - \int 3t^2 e^t dt\right]$$ ..... again by parts

$$\displaystyle =  t^4 e^t - 4\left[t^3e^t - 3 \int t^2 e^t dt\right]$$

$$\displaystyle = t^4e^t -4 t^3e^t + 12\left[t^2\int e^t dt - \int 2t e^t dt\right]$$ .... again by parts

$$ \displaystyle = t^4 e^t - 4t^3e^t+ 12 \left[ t^2e^t - 2 \int te^tdt]\right]$$

$$\displaystyle =t^4e^t - 4t63e^t + 12 \left[t^2e^t - 2 (te^t - \int e^tdt)\right]$$ .... again by parts

$$\displaystyle= t^4e^t - 4t^3e^t + 12t^2 e^t - 24te^t + 24e^t + C$$

Put the value of $$t$$

$$= (ln\, x)^4 x- 4 (ln\, x)^3 x+ 12 (ln \,x)^2 x - 24(ln\, x)x + 24x + C$$
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Subjective Medium Published on 17th 09, 2020
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