## How to solve indefinite integral of one divided by x rise to four by dx ?

Short answer: indefinite integral of one divided by x rise to four by dx is (-1/3x^3)+C. ∫ (1/x^4)dx is not particulary hard integral. You can solve it in 6 easy steps. We will walk you through and explain everything. Let's start.

## Table of contents

- Required assumtions
- Step by step solution of ∫ (1/x^4)dx
- What is indefinite integral of one divided by x rise to four by dx?
- Full video how to solve ∫ (1/x^4)dx
- Dictionary
- Want to solve next one?

## 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

## Step by step solution of ∫ (1/x^4)dx

We will solve ∫ (1/x^4)dx in **6** 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 1 divided by u is equal to u rise to -1. In our case u is equal to x.

### Step 2

We may swap 1 divided by x rise to four with x rise to minus four. Then we may directly use the formula, which says integral of u rise to n by du is equal to u rise to n+1 divided by n+1. In our case n is equal to -4.

### Step 3

Thus, we have x rise to minus 4 plus 1 divided by minus four plus one. 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, we have x rise to -3 divided by -3. To make it smoother we can modify further the equation by moving minus from denominator to the numerator. It will not affect our equation.

### Step 5

We can express it as -1 multiplied by x rise to -3 as there is no difference if we have minus next to x rise to -3 or we change it into -1 as it is not changing our final result. Also, we may get rid of minus in x rise to -3. Remembering that u rise to -n is equal to one divided by u rise to n. In our case u=x and n=3.

### Step 6

And we leave only -1 in numerator and move x rise to 3 to denominator.

## What is indefinite integral of one divided by x rise to four by dx?

We finally did it: ∫ (1/x^4)dx =(-1/3x^3)+C

## 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.