find the indefinite integral

Question

#1.    ∫(1+3s)^5 ds

 

#2. ∫7/x^10 dx.

 

#6. ∫e^-2y  dy

 

#10. ∫x^2+10x-12/x^4  dx

 

#12.∫1/5-2x dx

 

#13. ∫s^6 √(3+s^7) ds

 

#16.∫(5x^3+6)^3 (15x^2) dx

Answer 1

1. ∫(1+3s)^5 ds = (1/18)(1 + 3s)⁶ + C

 

Let u = 1 + 3s, then du = 3 ds. 

 

∫(1+3s)^5 ds = (1/3)∫u⁵ du = (1/3)(1/6)u⁶ + C = (1/18)(1 + 3s)⁶ + C

 

 

2. ∫7/x^10 dx = 7∫x^-10 dx = (-7/9)x^-9 + C = (-7/9)(1/x⁹) + C

 

 

6. ∫e^-2y dy = (-1/2)e^-2y + C

 

 

10. ∫x^2+10x-12/x^4  dx = (1/3)x³ + 5x² + 4/x³ + C

 

 

12. ∫1/5-2x dx = x/5 - x² + C

 

 

 
13. ∫s⁶ √(3+s⁷) ds = (1/7)∫√u du = (1/7)(2/3)u^3/2 + C = (2/21)(3 + s⁷)^3/2 + C

 

with u = 3 + s⁷ and du = 7s⁶ ds.

 

 

 

16. ∫(5x³ +6)³ (15x²) dx = ∫u³ du = (1/4)u⁴ + C = (1/4)(5x³ + 6)⁴ + C

 

with u = 5x³ + 6 and du = 15x² dx.