Integration of Fourth Root of x | 4√x Integration

The integration of fourth root of x is 4x^5/4/5 + C. In this post, we will find the integral of the fourth root of x by the power rule of integration.

Let us now learn how to integrate fourth root x.

Integral of Fourth Root of x

The fourth root of x is expressed as follows: $\sqrt[4]{x}$ which can be further rewrite by the rule of indices as $x^{\frac{1}{4}}$.

That is, $\sqrt[4]{x}=x^{\frac{1}{4}}$ …(I)

As fourth root of x is a power of x, we can use the power rule of integration to find its integral. By the power rule of integration, we have

∫xⁿ dx = xⁿ⁺¹/(n+1) + C where C is a constant of integration.

Putting n=1/4, we get the integral of fourth root of x as follows:

∫x^1/4 dx = ${\displaystyle \frac{x^{\frac{1}{4}+1}}{\frac{1}{4}+1}}$ + C

⇒ ∫$\sqrt[4]{x}$ dx = ${\displaystyle \frac{x^{5/4}}{5/4}}$ + C

⇒ ∫$\sqrt[4]{x}$ dx = ${\displaystyle \frac{4}{5}}{x}^{5/4}$ + C

So the integration of fourth root of x is 4x^5/4/5 + C where C denotes an integral constant, and this is proved by the power rule of integration.

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Question 1: Find the definite integral of fourth root of x from 0 to 1, that is Find ∫${}_0^1$x1/4 dx?

Solution:

From above we have that the integration of fourth root of x is 4x^5/4/5 + C. Therefore,

∫${}_0^1$ x^1/4 dx

= [4x^5/4/5 + C]${}_0^1$

= (4 ⋅ 1^5/4/5 + C) – (4 ⋅ 0^5/4/5 + C)

= 4/5+C – 0 -C

= 4/5.

So the integration of fourth root of x from 0 to 1 is equal to 4/5.

Question 2: What is the integration of 4?

Solution:

As 4 is a constant function of x, the integration of 4 with respect to x will be equal to

∫ 4 dx

= 4∫dx

= 4x + C.

So the integration of 4 is equal to 4x+C.

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