Evaluate: ∫dx/cos x(5 - 4sinx) ​

Question

Evaluate:

\(\int\frac{dx}{cosx(5-4sinx)}\)

∫dx/cos x(5 - 4sinx)

Answer 1

Let I = \(\int\frac{dx}{cosx(5-4sinx)}\)

Put t = sinx 

dt = cosxdx

A(1 + t)(5 - 4t) + B(1 - t)(5 - 4t) + C(1 + t)(1 - t) = 1 

Now Putting 1 + t = 0 

t = -1 

A(0) + B(2)(9) + C(0) = 1

B = 1/18

Now Putting 1-t = 0 

t = 1 

A(2) + B(0) + C(0) =1

A = 1/2

Now Putting 5 - 4t = 0

t = 5/4

A(0) + B(0) + C (1 - 25/16) = 1

C = -16/9

From equation(1),we get,