Stacie Worsley

2022-01-03

Determine the following indefinite integral.

$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx$

Jim Hunt

Beginner2022-01-04Added 45 answers

Step 1

Indefinite integral is basically an integral without upper and lower bounds, i.e. its boundaries are not set

Step 2

To find:$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx$

Now, taking$x}^{2$ common from the numerator and the denominator and simplifying:

$\int \frac{1}{{x}^{2}+1}dx$

Let$x=\mathrm{tan}u$

$dx={\mathrm{sec}}^{2}udu$

Substitute

$=\int \frac{1}{1+{\mathrm{tan}}^{2}u}\cdot {\mathrm{sec}}^{2}udu$

Now, as we know that$1+{\mathrm{tan}}^{2}u={\mathrm{sec}}^{2}u$

$=\int \frac{1}{{\mathrm{sec}}^{2}u}\cdot {\mathrm{sec}}^{2}udu$

$=\int 1\cdot du$

Integrating,

=u+c

Now, as$x=\mathrm{tan}u$

$u={\mathrm{tan}}^{-1}x$

Substituting value of u

$={\mathrm{tan}}^{-1}x+c$

$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx={\mathrm{tan}}^{-1}x+c$

Indefinite integral is basically an integral without upper and lower bounds, i.e. its boundaries are not set

Step 2

To find:

Now, taking

Let

Substitute

Now, as we know that

Integrating,

=u+c

Now, as

Substituting value of u

alkaholikd9

Beginner2022-01-05Added 37 answers

We need to evaluate the indefinite integral

$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx$

To do this we can use following rules,

$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C\Rightarrow \left(1\right)$

and,

where C is a real constant. Then,

$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx=\int \frac{{x}^{2}}{{x}^{2}({x}^{2}+1)}dx$

$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx=\int \frac{1}{{x}^{2}+1}dx$

$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx={\mathrm{tan}}^{-1}x+C$

where, C is the constant of integration.

Result:

The value of the given indefinite integral is,

$\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}}dx={\mathrm{tan}}^{-1}x+C$

where C is the constant of integration.

To do this we can use following rules,

and,

where C is a real constant. Then,

where, C is the constant of integration.

Result:

The value of the given indefinite integral is,

where C is the constant of integration.

karton

Expert2022-01-11Added 613 answers

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.