Question

★ Solve the following integral !!

$\large{\displaystyle{\sf \int \dfrac{x}{\sqrt[6]{x + 1}}dx}}$

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Answer 1

SOLUTION

We have,

$\large{\displaystyle{\sf l \: = \int \dfrac{x}{\sqrt[6]{x + 1}}dx}}$

The above integral can be solved by using u substitution

Now,

Assume,

$\sf \: \: u = x + 1$

Taking derivative both sides,we get :

$\leadsto \: \boxed{ \boxed{\sf \: du = dx} }$

The integral can be rewritten as :

$\displaystyle{\sf l \: = \int \dfrac{u - 1}{\sqrt[6]{u}} \: du}\\ \\ \longrightarrow \: \displaystyle{ \sf \: l = \int \: (u - 1) {u}^{ \frac{1}{6} }.du } \\ \\ \longrightarrow \: \displaystyle{ \sf \: l = \int \: (u {}^{ \frac{7}{6} - } u {}^{ \frac{1}{6} }).du }$

Applying the formula of integration,

$\longrightarrow \: \displaystyle{ \sf l = \: \dfrac{u {}^{ \frac{7}{6} + 1 }}{ \frac{7}{6} + 1} - \dfrac{ {u}^{ \frac{1}{6} + 1} }{ \frac{1}{6} + 1 } + c } \\ \\ \longrightarrow \: \sf \: l = \dfrac{6 {u}^{ \frac{13}{6} } }{13} - \dfrac{6 {u}^{ \frac{7}{6} } }{7} + c \\ \\ \longrightarrow \: \boxed{ \boxed{\sf \: l = \frac{6( \sqrt[6]{x + 1} ) {}^{13} }{7} - \frac{6( \sqrt[6]{x + 1}) {}^{7} }{7} + c}}$

NoTE

Integration :

• $\displaystyle{\sf I = \int \dfrac{x^{n+1}}{n + 1} }$

• C,known as Arbitrary Constant represents the family of integral solutions for a given integral

Integration can be solved by :

• Substitution Method
• Partial Fractions Method
• Inspection Method

Answer 2

INTEGRATING THE FOLLOWING EXPRESSION :-

$\displaystyle{\sf{\int \frac{x}{\sqrt[6]{x+1} } dx }}$

Let (x + 1) = a, then da/dx = 1 ⇒ da = dx  (Derivative of x + 1 is 1)

$\displaystyle{\sf{=\int \frac{a-1}{\sqrt[6]{a} } da}}$

$\displaystyle{\sf{=\int \bigg(a^\frac{5}{6}-\frac{1}{\sqrt[6]{a} } \bigg)da }}\\\\\displaystyle{\sf{=\int a^\frac{5}{6}da-\int \frac{1}{\sqrt[6]{a} }da }}$

_______________

$\displaystyle{\sf{solving\ \int a^\frac{6}{5} da\ :- }}$

By power rule,

$\displaystyle{\sf{=\int \frac{a^{\frac{5}{6}+1}}{\frac{5}{6} +1} }}\\\\\displaystyle{\sf{= \frac{6a^\frac{11}{6} }{11} }}$

_______________

$\displaystyle{\sf{Solving\ \int \frac{1}{\sqrt[6]{a} }da\ :- }}$

By power rule,

$\displaystyle{\sf{=\frac{6a^\frac{6}{5} }{5} }}$

_______________

Now, putting the values :-

$\displaystyle{\sf{=\int a^\frac{5}{6}da-\int \frac{1}{\sqrt[6]{a} }da }}\\\\\displaystyle{\sf{= \frac{6a^\frac{11}{6} }{11} -\frac{6a^\frac{5}{6} }{5} }}\\\\\displaystyle{\sf{= \frac{6(x+1)^\frac{11}{6} }{11} -\frac{6(x+1)^\frac{5}{6} }{5} }}$

At last,

$\displaystyle{\sf{= \frac{6(x+1)^\frac{11}{6} }{11} -\frac{6(x+1)^\frac{5}{6} }{5}+C }}$

$\boxed{\boxed{\sf{= \dfrac{\left(x+1\right)^\frac{5}{6}\left(30x-36\right)}{55}+C}}}$

$\rule{180}2$

Thanks  :-)