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Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx},\text{ n}\in\text{N},\text{n}\leq2$
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Answer

Let $\text{I}=\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx}\ ...(\text{i})$
Then,
$\text{I}=\int\limits^{\text{b}}_{\text{a}}\frac{\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}{\big(\text{a}+\text{b}-\text{x}\big)^\frac{1}{\text{n}}+\big[\text{a}+\text{b}-\big(\text{a}+\text{b}-\text{x}\big)\big]^{\frac{1}{\text{n}}}}\text{ dx}$ $\Bigg[\int\limits^{\text{b}}_{\text{a}}\text{f(x)}\text{dx}=\int\limits^{\text{b}}_{\text{a}}\text{f}(\text{a}+\text{b}-\text{x})\text{dx}\Bigg]$
$=\int\limits^{\text{b}}_{\text{a}}\frac{\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}{\big(\text{a}+\text{b}-\text{x}\big)^\frac{1}{\text{n}}+\text{x}^{\frac{1}{\text{n}}}}\text{ dx}$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{^{\frac{1}{\text{n}}}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx}$
$\Rightarrow2\text{I}=\int\limits^{\text{b}}_{\text{a}}\text{dx}$
$\Rightarrow2\text{I}=\big[\text{x}\big]^{\text{b}}_{\text{a}}=(\text{b}-\text{a})$
$\Rightarrow\text{I}=\frac{\text{b}-\text{a}}{2}$

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