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

Moderate

A

$3 {(1 + \sqrt[4]{x})}^{\frac{4}{3}} - \frac{12}{7} {(1 + \sqrt[4]{x})}^{\frac{7}{3}} + C$
B

$\frac{12}{7} {(1 + \sqrt[4]{x})}^{\frac{7}{3}} - 3 {(1 + \sqrt[4]{x})}^{\frac{4}{3}} + C$
C

$\frac{12}{7} {(1 + \sqrt[4]{x})}^{\frac{7}{3}} + 3 {(1 + \sqrt[4]{x})}^{\frac{4}{3}} + C$
D

$\frac{12}{7} {(1 + \sqrt[4]{x})}^{\frac{7}{3}} + \frac{1}{3} {(1 + \sqrt[4]{x})}^{\frac{4}{3}} + C$

Solution

$\int \frac{\sqrt[3]{1 + \sqrt[4]{x}}}{\sqrt{x}} d x$
$Let x = t^{4}$
$\begin{array}{l} d x = 4 t^{3} d t \\ = \int \frac{\sqrt[3]{1 + t}}{t^{2}} 4 t^{3} d t \\ = 4 \int \sqrt[3]{1 + t} t d t \end{array}$
$Put 1 + t = k^{3}$
$\begin{array}{l} d t = 3 k^{2} d k \\ = 3 k^{2} d k \\ = 4 \int k (k^{3} - 1) 3 k^{2} d k \\ = 12 \int k^{3} (k^{3} - 1) d k \\ = 12 \int (k^{6} - k^{3}) d k \\ = 12 [\frac{k^{7}}{7} - \frac{k^{4}}{4}] + C \\ = \frac{12}{7} k^{7} - \frac{12}{4} k^{4} + C \\ = \frac{12}{7} (1 + t)^{\frac{7}{3}} - 3 (1 + t)^{\frac{4}{3}} + C \end{array}$
$= \frac{12}{7} {(1 + \sqrt[4]{x})}^{\frac{7}{3}} - 3 {(1 + \sqrt[4]{x})}^{\frac{4}{3}} + C$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

SCAN ME