∫x31+x437dx is equal to

We have, I=∫x3⋅1+x437dx=34∫1+x4/37×43x1/3dx⇒ I=34∫1+x4/31/7d1+x4/3⇒ I=34×78×1+x4/38/7+C=21321+x4/38/7+C

∫x31+x437dx is equal to

We have, I=∫x3⋅1+x437dx=34∫1+x4/37×43x1/3dx⇒ I=34∫1+x4/31/7d1+x4/3⇒ I=34×78×1+x4/38/7+C=21321+x4/38/7+C

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$\int \sqrt[3]{x} \sqrt[7]{1 + \sqrt[3]{x^{4}}} d x$ is equal to

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none of these

$\frac{21}{32} {\{1 + \sqrt[3]{x^{4}}\}}^{8 / 7} + C$

$\frac{32}{21} {\{1 + \sqrt[3]{x^{4}}\}}^{8 / 7} + C$

$\frac{7}{32} {\{1 + \sqrt[3]{x^{4}}\}}^{8 / 7} + C$

answer is A.

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Detailed Solution

We have,

$\begin{array}{l} I = \int \sqrt[3]{x} \cdot \sqrt[7]{1 + \sqrt[3]{x^{4}}} d x = \frac{3}{4} \int \sqrt[7]{1 + x^{4 / 3}} \times \frac{4}{3} x^{1 / 3} d x \\ \Rightarrow I = \frac{3}{4} \int {(1 + x^{4 / 3})}^{1 / 7} d (1 + x^{4 / 3}) \\ \Rightarrow I = \frac{3}{4} \times \frac{7}{8} \times {(1 + x^{4 / 3})}^{8 / 7} + C = \frac{21}{32} {(1 + x^{4 / 3})}^{8 / 7} + C \end{array}$