Aof Panel

S

What is antiderivative of (y/2) e^yx+5 ?

(½) e^yx+5 + c

S

What is antiderivative of (5/2) e^5x+y

(½) e^5x+y + c

S

What is antiderivative of 6x² + 8x + (3/x) + 4 ?

2x³ + 4x² + 3lnx + 4x + c

S

What is antiderivative of x^1/3 ?

(¾) x^4/3+ c

S

What is antiderivative of x^1/2 with F(1) = 2 ?

(2/3) x^3/2+ c ; F(1) = 2 = (2/3) 1^3/2+ c = (2/3) + c ; c = 2 – (2/3) = 4/3

S

What is antiderivative of x^-1/2 with F(0) = -1 ?

2 x^1/2+ c ; F(0) = -1 = 2 0^1/2+ c = 0 + c ; c = -1

S

∫ ((3/x) + x) (x+1) dx = ?

∫ (3 + (3/x) + x² + x) dx = 3x + 3lnx + (1/3) x³ + (1/2) x² + c

S

∫ ((1 / x²) + 2x) (x + 3x²) dx = ?

∫ ((1/x) + 3 + 2x² + 6x³) dx] = lnx + 3x + (2/3) x3 + (3/2)x⁴ + c

S

∫ ((x + y - 2) / (x + y)^1/2) dx = ?

Change of variable : u = x + y ; du = dx ;

∫ ((u – 2) / (u^1/2)) du = ∫ ((u / (u^1/2)) – (2 / (u^1/2)) du = ∫ (u^1/2 – (2 / (u^1/2)) du

= (2/3) u^3/2 - 4 u^1/2 + c

= (2/3) (x + y)^3/2 - 4 (x + y)^1/2 + c

S

∫ ((yx + 2) / (yx + 1)^1/2) dx = ?

Change of variable : u = xy + 1 ; (1/y) du = dx ;

∫ ((u + 1) / (u^1/2)) du = ∫ [((u / (u^1/2)) + (1 / (u^1/2)) du = ∫ (u^1/2 + (1 / (u^1/2)) du

= (2/3) u^3/2 + 2 u^1/2 + c

= (2/3) (x + y)^3/2 + 2 (x + y)^1/2 + c

S

∫ x² (x³ + 1)⁵ dx = ?

Change of variable : u = x³ + 1 ; (1/3) du = x² dx ;

∫ (1/3) u⁵ du] = (1/18) u⁶ + c =

(1/18) (x³ + 1)⁶ + c

S

∫ x (yx² + 2)³ dx = ?

Change of variable : u = yx² + 2 ; (1/2y) du = x dx ;

∫ (1/2y) u³ du] = (1/8y) u⁴ + c =

(1/8y) (yx² + 2)⁴ + c

S

∫ ((3x² + 1) / (x³ + x + 1)) dx = ?

Change of variable : u = (x³ + x + 1) ; du = (3x²+ 1) dx ;

∫ (1/u) du = lnu + c

= ln(x³ + x + 1) + c

S

∫ (2x (2x² + 1) / (x⁴ + x² + 2)) dx = ?

Change of variable : u = (x⁴ + x² + 2) ; du = (4x³+ 2x) dx = 2x (2x²+ 1) dx ;

∫ (1/u) du = lnu + c =

ln(x⁴ + x² + 2) + c

S

∫ ((1/2) – x) e^-x dx = ?

Integration by parts : A = ∫ [u dv = uv - ∫ v du = uv – B ;

u = (1/2) – x ; du = - dx ;

dv = e^-x dx ; v = - e^-x ;

uv = ((1/2) – x) (- e^-x) ;

vdu = (- e^-x) (- dx) = e^-x dx ; B = - e^-x + c ;

A = ((1/2) – x) (- e^-x) - (- e^-x) + c = ((1/2) – x – 1) (- e^-x) + c

= ((1/2) + x) e^-x + c

S

∫ (x/y) e^2x dx = ?

Integration by parts : A = ∫ u dv = uv - ∫ v du = uv – B ;

u = x/y ; du = (1/y) dx ;

dv = e^2x dx ; v = (1/2) e^2x ;

uv = (x/y) (1/2) e^2x = (x/2y) e^2x ;

vdu = (1/2) e^2x (1/y) dx = (1/2y) e^2x dx ; B = (1/2y) (1/2) e^2x+ c ;

A = (x/2y) e^2x - (1/2y) (1/2) e^2x + c

= (1/2y) (x – (1/2)) e^2x + c

S

∫ x² ln(x/2) dx = ?

Integration by parts : A = ∫ u dv = uv - ∫ v du = uv – B ;

u = ln(x/2) ; du = (1/x) dx ;

dv = x² dx ; v = (1/3) x³ ;

uv = ln(x/2) (1/3) x³ ;

vdu = (1/3) x³ (1/x) dx = (1/3) x² dx ; B = (1/3) (1/3) x³+ c ;

A = ln(x/2) (1/3) x³ - (1/3) (1/3) x³ + c

= (1/3) x³(ln(x/2) – (1/3)) + c

S

∫ -x ln(yx) dx = ?

Integration by parts : A = ∫ u dv = uv - ∫ v du = uv – B ;

u = ln(yx) ; du = (1/x) dx ;

dv = -x dx ; v = (-1/2) x² ;

uv = ln(yx) (-1/2) x² ;

vdu = (-1/2) x² (1/x) dx = (-1/2) x dx ; B = (-1/2) (1/2) x²+ c ;

A = ln(yx) (-1/2) x² - (-1/2) (1/2) x² + c

= (-1/2) x²(ln(yx) – (1/2)) + c

S

∫ 5yx e^x/2 dx = ?

Integration by parts : A = ∫ u dv = uv - ∫ v du = uv – B ;

u = 5yx ; du = 5y dx ;

dv = e^x/2 dx ; v = 2 e^x/2 ;

uv = 5yx 2 e^x/2 = 10yx e^x/2 ;

vdu = 2 e^x/2 5y dx = 10y e^x/2 dx ; B = 20y e^x/2+ c ;

A = 10yx e^x/2 - 20y e^x/2 + c

= 10y (x – 2) e^x/2 + c

S

∫ (7yx + 9zx) e^5x dx = ?

Integration by parts : A = ∫ u dv = uv - ∫ v du = uv – B ;

(7yx + 9zx) = (7y + 9z) x = ax

u = ax ; du = a dx ;

dv = e^5x dx ; v = (1/5) e^5x ;

uv = ax (1/5) e^5x ;

vdu = (1/5) e^5x a dx ; B = (a/5) (1/5) e^5x+ c ;

A = ax (1/5) e^5x- (a/5) (1/5) e^5x + c = (a/5) (x – (1/5)) e^5x + c

= ((7y + 9z)/5) (x – (1/5)) e^5x + c

Indefinite Integral