Calculate definite integral of

dx/(x^4 * sqrt(x^2 + 3))
Over (1,3)

I start with the substitution x = sqrt(3)*tan t

so:
sqrt(x^2 + 3) = sqrt(3) * sec t
dx = sqrt(3) * sec^2 t dt
x^4 = 9 * tan^4 t

The integral simplifies to:
= dt/(tan^3 t * sin t)

How do I solve that?

Well, I call it:

(1/9)int cos^3 t dt/sin^4 t

(1/9) int (1-sin^2 t) cos t dt/sin^4 t

(1/9) int cos t dt/sin^4 t -(1/9) int cos t dt/sin^2 t

now integral of cos/sin^4 is of form 1/sin^3 (leaving the constants for you
and)
integral of cos/sin^2 is of form 1/sin (leaving the constants to you)

thanks damon! I follow perfectly.

To evaluate the integral ∫dx/(x^4 * sqrt(x^2 + 3)) over the interval (1,3), we made the substitution x = sqrt(3)*tan(t).

Let's continue solving the integral using the new variable t:

The substitution x = sqrt(3)*tan(t) gives us:
sqrt(x^2 + 3) = sqrt(3) * sec(t)
dx = sqrt(3) * sec^2(t) dt
x^4 = 9 * tan^4(t)

Substituting these values into the integral expression, we get:
∫ dt/(tan^3(t) * sin(t)) over the interval (1,3)

To simplify this expression further, we can use trigonometric identities.

Step 1: Rewrite tan^3(t) as (sin^3(t))/(cos^3(t)) and sin(t) as (sin^2(t))/(cos(t)).

∫ [dt/((sin^3(t))/(cos^3(t)) * (sin^2(t))/(cos(t)))] over the interval (1,3)

Step 2: Invert the division and multiply.
∫ [dt * cos^3(t) * cos(t)/(sin^3(t) * sin^2(t))] over the interval (1,3)

Step 3: Simplify.
∫ [dt * cos^4(t)/(sin^5(t))] over the interval (1,3)

Now, the integral expression is in a form that can be evaluated using basic integration rules.

To solve the integral, you can use either numerical methods or evaluate it analytically if possible. However, depending on the complexity of the function, it may not have a closed-form solution.

If you prefer to evaluate the integral analytically, you can try using integration techniques like substitution, integration by parts, or partial fractions to simplify and solve the expression.