How to solve integral ∫∛xdx

How to solve indefinite integral of cube root of x by dx ?

Short answer: indefinite integral of cube root of x by dx is (3/4)x∛x+C. ∫ ∛xdx is not particulary hard integral. You can solve it in 7 easy steps. We will walk you through and explain everything. Let's start.

Required assumtions
Step by step solution of ∫ ∛xdx
What is indefinite integral of cube root of x by dx?
Full video how to solve ∫ ∛xdx
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Required assumtions

Usually, we have some additional info about function f of (x). In our case:

f(x) belongs to real numbers
f(x) is integrable in that domain

assumtions for indefinite integral of cube root of x by dx

Step by step solution of ∫ ∛xdx

We will solve ∫ ∛xdx in 7 easy steps. Let's get started

Step 1

From symbol dx we know that differential of variable x indicates that the variable of integration is x. Let’s rewrite integrand - function inside the integral without fraction- knowing that n root of s is equal to s rise to 1 divided by n. In our case s is equal to x and n is equal to 3.

Step 2

Then we may directly use formula, which says integral of s rise to n by ds is equal to s rise to n+1 divided by n+1. In our case s is equal to x and n is equal to one-third.

Step 3

Thus, we have x rise to one third plus 1 divided by one third plus 1. We must also add constant C belonging to the set of real numbers because our solution is not a single function but a whole class of functions.

Step 4

So let’s just add 1 and one third in power and leave it for now as x rise to one and one third. In denominator we want to have the result of addition of 1 and one third written down as improper fraction. Thus, we have one third plus one and now we change 1 into fraction of three thirds as we want to have mutual denominator 3. We can now add in numerator 1 to 3 and we have 4 in numerator and 3 in denominator.

Step 5

Let’s come back now to numerator of our main problem we are solving. We have left x rise to one and one third and it’s time to smooth it a little. We know that s rise to n multiplied by s rise to m equals s rise to n plus m. In our case s is equal to x, n is 1 and m is one third. Thus, we may write down our numerator as x rise to 1 multiplied by x rise to one third. So x rise to 1 usually we just write down as x and then we can modify a little x rise to one third. s rise to one n-th equals n-th root of s and in our case s is equal to x and n is 3, then we have cube root of x here.

Step 6

Now we have our numerator smoothed and I believe it will be good to go further with denominator. If we have s divided by some constants a divided by b, we may write it as b divided by a multiplied by our s. So in our case s is equal to our numerator of main solution that is x times cube root of x , a is 4 and b is 3.

Step 7

And we have three fourths multiplied by x cube root of x plus constant C.

What is indefinite integral of cube root of x by dx?

We finally did it: ∫ ∛xdx=(3/4)x∛x+C

Full video of how to solve ∫ ∛xdx

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Dictionary

Integration (antidifferentiation)

Computation (process of finding) of an integral, opposite process to differentiation.

Integrand

Function placed between sign of integral and differential of variable of integration e.g. $${{ \int f(x)dx}}$$,
where:
$${{ \int }}$$- integration operator,
f(x) – integrand,
dx- differential of variable of integration x

Integrable function

Function that integral over its domain is finite.

Indefinite integral

Represents a class of primitive functions whose derivative is the integrand e.g. $${{\int f(x)dx=F(x)+C \Leftrightarrow F’(x)=f(x)}}$$
$${{C=const. }}$$,
$${{f, F,C \in R }}$$
R-real numbers
$${{\int f(x)dx}}$$ - indefinite integral of function f(x) by dx,
$${{F(x)+C}}$$ – a class of primitive functions that $${{F’(x)=f(x) }}$$,
F(x) - primitive function, usually written in capital letters,
R-real numbers.

Function

Function specified on a set X and having values in set Y is an assignment each element of set X specifically one element in set Y.
$${{f: X \rightarrow Y}}$$
f-function name,
X-set of elements of function f, domain of a function f
Y-set of function values of function f, codomain of a function f
$${{x\in X, y \in Y}}$$
$${{y=f(x) f: x\rightarrow y}}$$
$${{y=y(x), }}$$
y(x)-vales of the function named y,
x-independent variable,
y-dependent variable.