Evaluate the following integrals: _ 4 ^12x(x-4)^ 1 3 dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{4}^\limits{12}\text{x}(\text{x}-4)^{\frac{1}{3}}\text{dx}$

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Answer

Let $\text{I}=\int_{4}^\limits{12}\text{x}(\text{x}-4)^{\frac{1}{3}}\text{dx}$
Let $\text{x}-4=\text{t}$ Then, $\text{dx}=\text{dt}$
When $\text{x}=4,\text{t}=0$ and $\text{x}=12,\text{t}=8$
$\therefore\ \text{I}=\int\limits^8_0(\text{t}+4)\text{t}^{\frac{1}{3}}\text{dt}$
$\Rightarrow\text{I}=\int\limits^8_0\Big(\text{t}^{\frac{4}{3}}+4\text{t}^{\frac{1}{3}}\Big)\text{dt}$
$\Rightarrow\text{I}=\Big[\frac{3}{7}\text{t}^{\frac{7}{3}}+\frac{3}{1}\text{t}^{\frac{4}{3}}\Big]^8_0$
$\Rightarrow\text{I}=\frac{384}{7}+48$
$\Rightarrow\text{I}=\frac{720}{7}$

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