Evaluate the following integrals: ^ 7 _0 [3] x [3] x + [3] 7 -x dx

Let I= ^ 7 _0 [3] x [3] x + [3] 7-x dx ...(i) We know that ^ a _0f(x)= ^ a _0f(a-x) Hence, I= ^ 7 _0 [3] 7-x [3] 7-x + [3] x dx ...(ii) Adding (i) & (ii) 2I= ^ 7 _0 [3] x [3] x + [3] 7-x dx+ [3] 7-x [3] 7-x + [3] x dx 2I= ^ 7 _0 [3] x + [3] 7-x [3] x + [3] 7-x dx 2I= ^ 7 _0dx 2I= [x ]^7_0 I=7 2

Evaluate the following integrals: ^ 7 _0 [3] x [3] x + [3] 7 -x d…

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Question

Evaluate the following integrals:
$\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7}-\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}\text{ dx}\ ...(\text{i})$
We know that $\int\limits^{\text{a}}_0\text{f(x)}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})$
Hence,
$\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{7-\text{x}}}{\sqrt[3]{7-\text{x}}+\sqrt[3]{\text{x}}}\text{ dx}\ ...(\text{ii})$
Adding (i) & (ii)
$2\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}\text{ dx}+\frac{\sqrt[3]{7-\text{x}}}{\sqrt[3]{7-\text{x}}+\sqrt[3]{\text{x}}}\text{ dx}$
$2\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}\text{ dx}$
$2\text{I}=\int\limits^{7}_0\text{dx}$
$2\text{I}=\big[\text{x}\big]^7_0$
$\text{I}=\frac{7}{2}$

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