Question

Find the antiderivative of f(x)=cos2x/(sin^2x)*(cos^2x).

Accepted Answer

We have the function f(x) = cos 2x /(sin x)^2*(cos x)^2
cos 2x = (cos x)^2 - ( sin x)^2
=> cos 2x /(sin x)^2*(cos x)^2
=> [(cos x)^2 - ( sin x)^2] / (sin x)^2*(cos x)^2
=> 1/ (sin x)^2 - 1/ (cos x)^2
=> (cosec x)^2 - (sec x)^2
So, Int [ f(x)]
=> Int [ (cosec x)^2 - (sec x)^2 dx]
=> Int [(cosec x)^2 dx] - Int [(sec x)^2 dx]
=> - tan x  - cot x + C
Therefore the required antiderivative of f(x)=cos2x/(sin^2x)*(cos^2x) = -tan x  - cot x + C

Answer

To find the anti derivative of  f(x)=cos2x/(sin^2x)*(cos^2x.
Solution:
We should  get Int f(x) dx  = Int cos2x/(sin^2x)*(cos^2x) dx
Int f(x) dx = Int {(cos^2x- sin^2x)/(cos^xsin^2x)} dx, cos2x = cos^2x- sin^2x is an identity.
Int f(x) dx = Int (1/sin^2x) dx - Int {1/cos^2x) dx.
f(x)dx = Int {cosec^2x}dx -  Int sec^2x dx dx.
f(x) =  -cot x - tanx  + C, as d/dx tan = sec^2x and d/dx cot  x= cosecx.
Therefore anti derivative of f(x) = cos2x/(sin^2x)*(cos^2x) =  -cot x - tanx  + C.

Answer

Of course, we can determine!
You did not specified what made you to be doubtful regarding the possibility to find the anti-derivative, but I think that the cause is that the argument of the function from numerator is different from the argument of the function from denominator.
We'll re-write the numerator of the double angle:
cos 2x= (cos x)^2 - (sin x)^2
We'll re-write the indefinite integral:
Int cos2x dx/(cos x)^2*(sin x)^2 = Int {[(cos x)^2 - (sin x)^2]dx/(cos x)^2*(sin x)^2}
We'll use the property of integrals to be additive:
Int cos2x dx/(cos x)^2*(sin x)^2 = Int (cos x)^2dx/(cos x)^2*(sin x)^2 - Int (sin x)^2]dx/(cos x)^2*(sin x)^2
We'll simplify and we'll get:
Int cos2x dx/(cos x)^2*(sin x)^2 = Int dx/*(sin x)^2 - Int dx/(cos x)^2
Int cos2x dx/(cos x)^2*(sin x)^2 = -cot x - tan x + C

Math

Find the antiderivative of f(x)=cos2x/(sin^2x)*(cos^2x).

Expert Answers

We’ll help your grades soar

Related Questions

Find sin 2x, cos 2x, and tan 2x from the given information.Find sin 2x, cos 2x, and tan 2x from the given information. sin x= 8/17 , x in Quadrant I sin 2x =____________ cos...

cos x-sin 3x-cos 2x = 0

Evaluate the integral of function y=cos2x/cos^2x*sin^2x.

If f(sin x) = cos 2x, find f(cos x).

Determine the expression of antiderivative of f(x)=1/cos^4(x)