Integration of cube root of x | Integral of cube root of x

The cube root of x is a number whose cube is x, in other words, (cube root of x)^3 = x. In this post, we will evaluate the integration of the cube root of x. To do so, we will use the power rule of derivatives which is given below:

$\displaystyle \int x^n dx = \dfrac{x^{n+1}}{n+1}+c$ $ \cdots (i)$

where $c$ is an integration constant.

Also Read: Integration of root x

What is the integration of cube root of x? Let’s find out.

Integration of Cube Root of x

Question: Find the integration of cube root x, that is, find

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

Answer: 

At first, by the rule of indices, we can write the cube root of $x$ as $x^{\frac{1}{3}}$. That is,

$\sqrt[3]{x}=x^{\frac{1}{3}}$

Using this fact, we have

$\displaystyle \int \sqrt[3]{x} dx$ $= \displaystyle\int x^{\frac{1}{3}} dx$

$=\dfrac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1}+c \quad$ by the above power rule (i) of derivatives.

$=\dfrac{x^{\frac{1}{3}+1}}{\frac{4}{3}}+c$

$=\dfrac{3}{4} x^{1+\frac{1}{3}}+c$

$=\dfrac{3}{4} x \cdot x^{\frac{1}{3}}+c$

$=\dfrac{3}{4} x \sqrt[3]{x}+c$

So the integration of cube root of x is equal to $\dfrac{3}{4} x$ ∛x. In other words,

$\displaystyle \int$ ∛x $dx=\dfrac{3}{4} x$ ∛x +c where $c$ is an integration constant.

Also Read:

Integration of x/(1+x2)	Integration of x/(1+x)
∫√(1+sin2x) dx	Integral of e^(x^2)
Integration of log(sinx)	∫xndx Formula, Proof

Integration of 1 by Cube Root x

Question: Find the integration of 1 by cube root x, that is, calculate

$\displaystyle \int \dfrac{1}{\sqrt[3]{x}} dx$

Answer: 

By the power rule of indices, we have  1/∛x $=1/x^{\frac{1}{3}}$. This can be further written as $x^{-1/3}$. Therefore, we have

$\displaystyle \int \dfrac{1}{\sqrt[3]{x}} dx$ $=\displaystyle \int x^{-\frac{1}{3}} dx$

Now applying the above power rule (i) of derivatives for $n=-1/3$, this is equal to

$=\displaystyle \int x^{-\frac{1}{3}} dx$

$=\dfrac{x^{-\frac{1}{3}+1}}{-\frac{1}{3}+1}+c$

$=\dfrac{x^{\frac{2}{3}}}{\frac{2}{3}}+c$

$=\dfrac{3}{2} x^{\frac{2}{3}}+c$

So the integration of 1 by cube root x is 3/2 x^2/3+c, where c is an integration constant.

Derivative of root x

Derivative of x sin x

Definite integration of Cube Root x

Now, we will find a definite integral involving the cube root of x.

Question: Evaluate the integration of cube root x from 0 to 1.

$\displaystyle \int_0^1 \sqrt[3]{x} dx$

Answer:

From the above we know that $\int \sqrt[3]{x} dx=\dfrac{3}{4} x \sqrt[3]{x}$.

So we have

$\displaystyle \int_0^1 \sqrt[3]{x} dx$ =$\left[\dfrac{3}{4} x \sqrt[3]{x} \right]_0^1$

$=\dfrac{3}{4}[x \sqrt[3]{x}]_0^1$

$=\dfrac{3}{4}[1-0]$

$=\dfrac{3}{4}$.

Have You Read These?

Derivative of log(3x)

Derivative of root(1+x)

Derivative of log(cos x)

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