find the following indefinite integrals

Question

Answer 1

(c) ʃ (cosx)⁴ sinxdx

cosx = t

Apply derivative each side.

dt = -sinx dx

= ʃ (t)⁴ (-dt)

Apply integral.

= - ((t)^{4 +1} )/(4+1) + c

= - t⁵/5 +c

Substitute cosx = t in the above equation.

= -cosx⁵/5

 The solution is = -cosx⁵/5

Answer 2

(b)  ∫ x*cos(x^2) dx

u = x^2 ⇒ du = 2x dx

= 1/2 ∫ cos(x^2) (2x dx)
= 1/2 ∫ cos(u) du

Apply integral.
= (1/2)sin(u) + C

Substitute u = x^2 in the above equation
= (1/2)sin(x^2) + C,
 

Answer 3

(a) ʃ dx / (x(ln x)^3)

t = ln x ⇒dt = 1/x dx

= ʃ (1/(t^3))du

Apply integral.

= 1/(- 2t^2) +c

Substitute t = ln x  in the above equation.

= - 1/(2(lnx^2))

The solution is = - 1/(2(lnx^2))
 

Answer 4

(d)

Apply integral.

Substitute t = sin^-1  in the above equation.