Question

integrate the following.​

Answer 1

$\implies \displaystyle \int \sqrt[3]{x} \: \: dx$

$\implies \displaystyle \int {x}^{ \frac{1}{3} } \: \: dx$

We know that :-

$\implies \displaystyle \int {x}^{ n} \: \: dx = \dfrac{ {x}^{n + 1} }{n + 1 } + c$

Therefore :-

$\implies \displaystyle \int {x}^{ \frac{1}{3} } \: \: dx = \dfrac{ {x}^{ \frac{1}{3} + 1} }{ \dfrac{1}{3} + 1 } + c$

$\implies \displaystyle \int {x}^{ \frac{1}{3} } \: \: dx = \dfrac{ {x}^{ \frac{1 + 3}{3}} }{ \dfrac{1 + 3}{3} } + c$

$\implies \displaystyle \int {x}^{ \frac{1}{3} } \: \: dx = \dfrac{ {x}^{ \frac{4}{3}} }{ \dfrac{4}{3} } + c$

$\implies \displaystyle \int {x}^{ \frac{1}{3} } \: \: dx = \dfrac{ 3{x}^{ \frac{4}{3}} }{ {4}} + c$

where c is constant

Answer :

$\implies \dfrac{ 3{x}^{ \frac{4}{3}} }{ {4}} + c$

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$\large\bf\red{Additional-Information}$

∫ 1 dx = x + C

∫ sin x dx = – cos x + C

∫ cos x dx = sin x + C

∫ sec2 dx = tan x + C

∫ csc2 dx = -cot x + C

∫ sec x (tan x) dx = sec x + C

∫ csc x ( cot x) dx = – csc x + C

∫ (1/x) dx = ln |x| + C

∫ ex dx = ex+ C

∫ ax dx = (ax/ln a) + C

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