f(x) = 5cos^2 x sec x - 4sin x csc x Integration & Derivative?

Question

Step By Step

Answer 1

Well first you need to simplify f(x)
Step 1;
f(x) = 5cos^2 x sec x - 4sin x csc x
{ secx = 1/cosx & cscx = 1/sinx }
f(x) = 5( cos^2 x (1/cosx ) -4( sinx (1/sinx ))
Now we can remove cosx and sinx in this Fraction

so the result would be 5( cosx) - 4 = 5cox - 4

Step 2 ;
Derivative;
if f(x)=5cox - 4
{ d/dx cosx = -sinx } and remmeber Derivative of a real number like '4' is always equal ' 0'
f '(x) = 5 * (-sinx) - 0
f '(x) = - 5 sinx

Step 3;
Intergral;
f(x)=5cox - 4

now lets back ;
∫f(x) = ∫5cox - 4 = ∫5cox- ∫4 = 5 (∫cosx ) - 4x
{ if ∫cos x dx = sin x + C }
so we have
∫f(x) = 5 ( sin x + C) - 4x = 5 sinx - 4x + c
Very Good Question. i like to answer your Question.

Answer 2

First simpify.
secx = 1/cosx, so
5cos^2 x secx = 5cos^2 x (1/cosx) = 5cosx
Also,
cscx = 1/sinx, so
4sinx cscx = 4sinx (1/sinx) = 4
So your problem simplifies to
f(x) = 5cosx - 4
Now the problem is much easier to work with.
The derivative is -5sinx
and the integral is 5sinx - 4x + C, where C is a constant.

Answer 3

First simplify: sec x=1/cos x and csc x=1/sin x. Thus:
f(x)=5 cos x - 4
Thus:

f'(x) = -5 sin x

∫f(x) dx = 5 sin x - 4x + C

Answer 4

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