How to solve integral ∫(∜(x^3)/x^4)dx

How to solve indefinite integral of 4th root of x cube divided by x rise to the 4th power by dx ?

Short answer: indefinite integral of 4th root of x cube divided by x rise to the 4th power by dx is (-4/(9x^2)∜x))+C. ∫ (∜(x^3)/x^4)dx is not particulary hard integral. You can solve it in 14 easy steps. We will walk you through and explain everything. Let's start.

Required assumtions
Step by step solution of ∫ (∜(x^3)/x^4)dx
What is indefinite integral of 4th root of x cube divided by x rise to the 4th power by dx?
Full video how to solve ∫ (∜(x^3)/x^4)dx
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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 4th root of x cube divided by x rise to the 4th power by dx

Step by step solution of ∫ (∜(x^3)/x^4)dx

We will solve ∫ (∜(x^3)/x^4)dx in 14 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. We may apply formula which says n-th root of s equals s rise to one divided by n. In our case s is x cube and n is 4.

Step 2

And we have in denominator x cube rise to one-quarter. Now we may take advantage of the formula s rise to n and that rise to m equals s rise to n multiplied by m. In our case s is x, n is 3 and m is one-quarter.

Step 3

In that way we have x rise to three quarters divided by x rise to 4.

Step 4

Now we may take advantage of the formula s rise to n divided by s rise to m equals s rise to n minus m. In our case s equals x, n is three-quarters and m is 4..

Step 5

And we have x rise to three-quarters minus 4. Let’s leave three-quarters and now we can write 4 as improper fraction 16 divided by 4. We have common denominator 4 and in numerator 3 minus 16 equals -13.

Step 6

We can insert it to our main solution, and we have x rise to -13 divided by four.

Step 7

So, let’s come back to the fact that we were modifying integrand and what we have to solve is the whole integral. But in current form we may make use of formula for solving integral of s rise to n which is s rise to n plus 1 divided by n+1. We also have to add constant C to make sure that we have all our solutions taken into account as our solution make whole class of function not a single one. And now in our case s is equal to x and n is minus 13 divided by 4.

Step 8

We’ve got x rise to minus thirteen quarters plus 1 divided by minus thirteen quarters plus one and plus constant C. Let’s leave minus thirteen quarters and now we can write 1 as improper fraction 4 quarters. We have common denominator 4 and in numerator minus 13 plus 4. That give us -9 divided by 4 and we can now insert it to our main solution.

Step 9

We’ve got x rise to -9 quarters divided by -9 quarters plus constant C.

Step 10

We can smooth it a little taking advantage of formula which says that s divided by a divided by b equals b divided by a multiplied by s. In our case s is equal to x rise to minus nine quarters, a is -9, b is 4.

Step 11

And we have -4 divided by 9 multiplied by x rise to minus 9 quarters plus C. We can rewrite also x rise to minus 9 quarters knowing that s rise to minus n equals 1 divided by s rise to n. In our case s is x and n is 9 quarters.

Step 12

Our solution now is minus 4 divided by 9 multiplied by 1 divided by x rise to nine quarters plus C.

Step 13

Let’s modify a little x rise to 9 quarters. We may express it as x rise to 8 plus 1 and then focus on power of x . We may have there 8 quarters plus one quarter and then we may write the whole expression as x square multiplied by x rise to one-quarter and further as x square and one/quarter as 4th root of x.

Step 14

In result we have in main solution -4 in numerator and 9 multiplied by x square multiplied by 4th root of x in denominator plus constant C.

What is indefinite integral of 4th root of x cube divided by x rise to the 4th power by dx?

We finally did it: ∫ (∜(x^3)/x^4)dx=(-4/(9x^2)∜x))+C

Full video of how to solve ∫ (∜(x^3)/x^4)dx

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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.