evaluate the integral

Question

#17. ∫(8x^2-7x-3)dx

 

#22.∫e^14x dx

 

#24. ∫8x^7+5x^4/x^8+X^5  dx.

 

#25.∫3x^5 dx

 

#28. ∫(9+x^3/2) dx

 

find the indefinite integral

#20. ∫10x/√16-3x^2 dx

 

#27. ∫4x^2/7-10x^3 dx

Answer 1

Jenny,

 

Since these are indefinite integrals that have no "from-to" integration limits on the variable x, the answers will be expressed in terms of x.  If limits were specified, then the answer s would just be numerical values.

 

#17. ∫(8x^2-7x-3)dx = ∫8x²dx - ∫7xdx - ∫3dx = (8/3)x³ -(7/2)x² -3x.

#22.∫e^14x dx = (1/14)e^(14x).

#24. ∫8x^7+5x^4/x^8+X^5 dx = ∫8x⁷dx + ∫5(x4/x8)dx + ∫x⁵ dx = (8/8)x⁸ + ∫5(x^-4)dx + (1/6)x⁶ 

                                                = x⁸ + 5(-1/5)x^-5 + (1/6)x⁶  =  x⁸ + (1/6)x⁶ -x^-5.

#25.∫3x^5 dx = (3/6)x⁶ = (1/2)x⁶.

#28. ∫(9+x^3/2) dx = 9x + (2/5)x^(5/2).

#20. ∫10x/√16-3x^2 dx = (10/-6][16 - 3x²]^(1/2) = - (5/3) [16 - 3x²]^(1/2) .

#27. ∫4x^2/7-10x^3 dx = 4(7/9)x^(9/7) - 10(1/4)x⁴ = (28/9)x^(9/7) - (5/2)x⁴.

 

Answer 2

17. (8/3)x³ - (7/2)x² - 3x + C

 

   ∫axⁿ dx = (a/(n+1))xⁿ⁺¹ + C

 

 

22. (1/14)e^14x + C    

 

   ∫e^ax dx = (1/a)e^ax + C

 

 

24. ln|x⁸ + x⁵| + C

 

   ∫du/u = ln|u| + C.  Here u = x⁸ + x⁵ and du = (8x⁷ + 5x⁴) dx.

 

25. (1/2)x⁶ + C

 

   Same strategy as in #17.

 

 

28. 9x + (2/5)x^5/2 + C

 

   Same strategy as in #17.

 

 

20. (-10/3)√(16 - 3x²) + C

 

   Use u = 16 - 3x². Then, du = -6x dx.

 

   ∫(10x/√(16 - 3x²) dx = (-5/3)∫du/√u = (-5/3)(2)√u + C

 

 

27. (-2/15)ln|7 - 10x³| + C

 

   Use u = 7 - 10x³.  Then, du = -30x² dx

 

   ∫4x²/(7 - 10x³) dx = (-4/30)∫du/u = (-2/15)ln|u| + C

 

 

Check by differentiation.