Home > Calculus > Definite and indefinite integrals

How do you find the indefinite integral of #int (x+1/x)^2dx#?

Answer 1

Emilia Aguilar

#int(x+1/x)^2dx=x^3/3+2x-1/x+C#

Note that:

#(x+1/x)^2=(x+1/x)(x+1/x)#

#color(white)((x+1/x)^2)=x^2+2x(1/x)+(1/x)^2#

#color(white)((x+1/x)^2)=x^2+2+x^-2#

So:

#I=int(x+1/x)^2dx=intx^2dx+2intdx+intx^-2dx#

Using #intdx=x# and #intx^ndx=x^(n+1)/(n+1)# we obtain:

#I=x^3/3+2x+x^-1/(-1)+C#

#I=x^3/3+2x-1/x+C#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Isaiah Allen

To find the indefinite integral of (\int (x + \frac{1}{x})^2 , dx), first expand the integrand:

[ \int \left(x + \frac{1}{x}\right)^2 dx = \int \left(x^2 + 2\cdot x \cdot \frac{1}{x} + \frac{1}{x^2}\right) dx = \int \left(x^2 + 2 + \frac{1}{x^2}\right) dx ]

Now, integrate each term separately:

(\int x^2 , dx = \frac{x^3}{3} + C_1)
(\int 2 , dx = 2x + C_2)
(\int \frac{1}{x^2} , dx = \int x^{-2} , dx = -x^{-1} + C_3 = -\frac{1}{x} + C_3)

Combine these results and include a single constant of integration:

[ \int \left(x + \frac{1}{x}\right)^2 dx = \frac{x^3}{3} + 2x - \frac{1}{x} + C ]

Where (C = C_1 + C_2 + C_3) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.