Question

integral x+5upon 3xsquare +13x-10

Answer 1

Answer:

Step-by-step explanation

(x+5)/(3x^2+13x-10)

∫(x+5)/(3x^2+13x−10)dx

Simplify the integrand:

∫(x+5)/(3x^2+13x−10)dx=∫1/(3x−2)dx

Let u=3x−2.

Then du=(3x−2)′dx=3dx and we have that dx=du/3

Thus

∫1/(3x−2)dx=∫1/3u du

Apply the constant multiple rule ∫cf(u)du=c∫f(u)du

with c=1/3

and f(u)=1/u

∫1/3u du=(1/3∫1/u du)

The integral of 1/u is

ln(|u|)/3

∫1/udu=ln(|u|)/3

Recall that u=3x−2:

ln(|u|)/3=1/3.ln(|(3x−2)|)

Therefore,

∫(x+5)/(3x2+13x−10)dx=1/3.ln(|3x−2|)

Add the constant of integration:

∫(x+5)/(3x2+13x−10)dx=1/3.ln(|3x−2|)+C

Answer: ∫(x+5)/(3x2+13x−10)dx=1/3.ln(|3x−2|)+C