Compute the following indefinite integrals. Please label things carefully and show all your work!

Question

Compute the following indefinite integrals. Please label things carefully and show all your work!

(a) ∫(5x² - 3sin(x) + √3) dx

(b) (( ⁵√ t²)/(t) + 2cos(t)) dt

(c) (-7y-3)/(³√7y²+ 6y) dy

Answer 1

a) ∫5x²-3sin(x)+√3 dx = 5∫x²dx-3∫sin(x)dx+√3∫dx = (5/3)x³+3cos(x)+(√3)x + C

b) ∫ ⁵√t²/t + 2cos(t) dt = ∫ t^2/5/t dt + 2∫cos(t)dt = ∫t^-3/5dt + 2∫cos(t)dt =(5/2)t^2/5 + 2sin(t) + C

c) let u = 7y²+6y du = (14y+6)dy = 2(7y+3) dy

(-1/2)∫(7y+3)/(7y+3) u^-1/3 du = (-1/2)∫u^-1/3du = (-1/2)(3/2)u^2/3 + C = (-3/4)(7y²+6y)^2/3 + C