Integrals Test - 7

I = ∫√(x ²+ 5x)dx

= ∫√(x ²+ 5x + 25/4 - 25/4)

= ∫√{(x + 5/2)²- (5/2)²}

={1/2(x+5/2)(√x ²+ 5x)} - {25/8 log{(x + 5/2)+√x ²+ 5x}}

= {(2x + 5)/4 (√x ²+ 5x)} - {25/8 log{(x + 5/2)+√x ²+ 5x}}

Thus, option D is correct...

sin² x = 1 - cos² x
∫sinx(sin² x - 3cos² x + 15)dx
Put cos² x = t
 ∫sinx(1 - cos² x - 3cos² x + 15)dx
=  ∫sinx (16 - 4cos² x)dx
Put t = cosx, differentiate with respect to x, we get  
dt/dx = -sinx
= -  ∫[(16 - 4t² )]^1/2 dt
= -2 ∫[(2)² - (t)² ]^½
= -2{[(2)² - (t)² ]^½  + 2sin^-1 (t/2)} + c
= - cosx {[4 - (cos)² x]^½ - 4sin^-1 (cosx/2)} + c

 (x)^½ (a - x)^½ dx
=  ∫(ax - x² )^½ dx
=  ∫{-(x² - ax)^½ } dx
=  ∫{-(x² - ax + a^2/4 - a^2/4 )^½ } dx
=  ∫{-(x - a/2)² - a^2/4 } dx
=  ∫{(a/2)² - (x - a/2)² } dx
=  ½(x - a/2) {(a/2)² - (x - a/2)² } + (a^2/4 ) (½sin^-1 (x - a)/a² )
= {(2x - a)/4 (ax - x² )^½ } + {a^2/8 sin^-1 (2x - a)/a} + c

None

Apply Trig Substitution: x=4sin(u)​

Let y = e^x
dy/dx = e^x dx
= ∫(9 - y² )^½dy
= ∫[(3)² - (y)² ]^½ dy
Apply [(a)² - (x)² ] formula
⇒y/2[(3)² - (y)² ]^½ + [(3)² ]/2 sin^-1 (y/3) + c
= e^x /2[(3)² - (y)² ]^½ + [9]/2 sin^-1 (e^x /3) + c

= π/ 4