integral(sin(2x)+5tan^2 x csc^2 x)dx=
in the first term, let u = 2x
Int(sin(2x)) = 1/2 Int(sin(u) du)
in the second term, let u = cot(x), so du = - csc^2 x dx
int(5tan^2 x csc^2 x) = -5 Int(1/u^2 du), a simple power rule term
To solve the integral of sin(2x) + 5tan^2x csc^2x, we can break it down into two separate integrals and then find the sum of their solutions.
First, let's solve the integral of sin(2x).
∫ sin(2x) dx
To integrate sin(2x), we can use the substitution method. Let's substitute u = 2x, which implies du = 2dx. Rearranging this equation gives dx = du/2. Now we substitute:
∫ sin(2x) dx = ∫ sin(u) (du/2)
Since the integral of sin(u) is -cos(u), we can continue:
∫ sin(2x) dx = ∫ sin(u) (du/2) = -(1/2)∫ cos(u) du = -(1/2)sin(u) + C
Substituting back in for u gives:
∫ sin(2x) dx = -(1/2)sin(2x) + C1
Now let's solve the integral of 5tan^2x csc^2x.
∫ 5tan^2x csc^2x dx
The integral of tan^2x csc^2x can be solved using the substitution method as well. Let's substitute u = tan(x), which implies du = sec^2(x) dx. Rearranging this equation gives dx = du / sec^2(x). Now we substitute:
∫ 5tan^2x csc^2x dx = ∫ 5u^2 (du / sec^2(x))
Simplifying further:
∫ 5tan^2x csc^2x dx = ∫ 5u^2 cos^2(x) du
Using the trigonometric identity, cos^2(x) = 1 - sin^2(x), we can substitute for cos^2(x):
∫ 5tan^2x csc^2x dx = ∫ 5u^2 (1 - sin^2(x)) du
Expanding further:
∫ 5tan^2x csc^2x dx = 5∫ u^2 du - 5∫ u^2 sin^2(x) du
The first term can be solved straightforwardly:
5∫ u^2 du = (5/3)u^3 + C2
For the second term, let's solve it separately.
-5∫ u^2 sin^2(x) du
Since sin^2(x) can be expressed as (1 - cos^2(x)), we substitute:
-5∫ u^2 (1 - cos^2(x)) du = -5∫ u^2 du + 5∫ u^2 cos^2(x) du
The first term is the same as what we already solved:
-5∫ u^2 du = -(5/3)u^3 + C3
For the second term, we already know its solution:
5∫ u^2 cos^2(x) du = 5(1/3)u^3 + C4 = (5/3)u^3 + C4
Combining the results:
-5∫ u^2 sin^2(x) du = -(5/3)u^3 + C3 + (5/3)u^3 + C4 = (5/3)u^3 + C5
Now let's substitute back in for u:
-5∫ u^2 sin^2(x) du = (5/3)tan^3(x) + C5
Finally, we can find the integral solution by adding up the solutions we obtained:
∫ sin(2x) + 5tan^2x csc^2x dx = -(1/2)sin(2x) + (5/3)tan^3(x) + C
Therefore, the integral of sin(2x) + 5tan^2x csc^2x with respect to x is equal to -(1/2)sin(2x) + (5/3)tan^3(x) + C, where C represents the constant of integration.