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.
