Evaluate the following integrals: ^ 5 _0 [4] x+4 [4] x+4 + [9] 9-… - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{5}_0\frac{\sqrt[4]{\text{x}+4}}{\sqrt[4]{\text{x}+4}+\sqrt[9]{9-\text{x}}}\text{ dx}$

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Answer

Let $\text{I}=\int\limits^{5}_0\frac{\sqrt[4]{\text{x}+4}}{\sqrt[4]{\text{x}+4}-\sqrt[9]{9-\text{x}}}\text{ dx}\ ....(\text{i})$
$\text{I}=\int\limits^{5}_0\frac{\sqrt[4]{9-\text{x}}}{\sqrt[4]{9-\text{x}}-\sqrt[4]{\text{x}+4}}\text{ dx}$
$\Big[\text{Using},\int\limits^{\text{a}}_0\text{f(x)}\text{dx}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})\text{dx}\Big]$
$\text{I}=\int\limits^{5}_0\frac{\sqrt[4]{9-\text{x}}}{\sqrt[4]{\text{x}+4}-\sqrt[4]{9-\text{x}}}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii)
$2\text{I}=\int\limits^{5}_0\frac{\sqrt[4]{\text{x}+4}}{\sqrt[4]{\text{x}+4}-\sqrt[4]{9-\text{x}}}-\frac{\sqrt[4]{9-\text{x}}}{\sqrt[4]{\text{x}+4}-\sqrt[4]{9-\text{x}}}\text{ dx}$
$=\int\limits^{5}_0\frac{\sqrt[4]{\text{x}+4}-\sqrt[4]{9-\text{x}}}{\sqrt[4]{\text{x}+4}-\sqrt[4]{9-\text{x}}}\text{ dx}$
$=\int\limits^{5}_0\text{dx}$
$=\big[\text{x}\big]^5_0$
$=5$
Hence, $\text{I}=\frac{5}{2}$

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