What is the best substitution to make to evaluate the integral of the quotient of cosine of 2 times x and the square root of the quantity 5 minus 2 times the sine of 2 times x, dx?
u = sin(2x)
u = cos(2x)
u = 2x
u = 5 − 2sin(2x)
To evaluate the integral of the quotient of cosine of 2x and the square root of 5 - 2sin(2x), we need to find the best substitution that simplifies the integral.
Let's consider the given options for substitution: u = sin(2x), u = cos(2x), u = 2x, and u = 5 - 2sin(2x).
First, let's examine the given function. We have cosine of 2x in the numerator and the square root of (5 - 2sin(2x)) in the denominator.
Out of the given options, the most suitable substitution would be u = sin(2x).
Here's why:
If we substitute u = sin(2x), the derivative du/dx can be found by differentiating both sides:
du/dx = d(sin(2x))/dx
du/dx = 2cos(2x)
Moreover, we can express cos(2x) in terms of u using the Pythagorean identity:
cos^2(2x) + sin^2(2x) = 1
1 - sin^2(2x) + sin^2(2x) = 1
cos^2(2x) = 1 - sin^2(2x)
cos^2(2x) = 1 - u^2
cos(2x) = sqrt(1 - u^2)
Substituting cos(2x) and du/dx back into the original expression, we get:
Integral [cos(2x) / sqrt(5 - 2sin(2x))] dx
= Integral [(sqrt(1 - u^2)) / sqrt(5 - 2u)] * (du / 2cos(2x))
= (1/2) * Integral [(sqrt(1 - u^2)) / sqrt(5 - 2u)] du
Now, we have transformed the integral into a simpler form using the substitution u = sin(2x). We can proceed to evaluate this integral by using algebraic manipulations, partial fractions, or using trigonometric identities to simplify the expression further.
Please Note: The process of evaluating the integral might involve additional steps depending on the complexity of the function.
To evaluate the integral of the given expression, we can make the substitution:
u = sin(2x)
This is the best substitution to use because it will simplify the expression and allow us to easily integrate.
Now, let's find the value of du in terms of dx:
du = 2cos(2x)dx
To convert the original integral in terms of u, we need to rewrite the expression in terms of u:
cos(2x) = 1 - 2sin^2(x)
The denominator becomes:
√(5 - 2sin(2x)) = √(5 - 2u)
Now, let's substitute these values into the integral:
∫ (cos(2x) / √(5 - 2sin(2x))) dx
= ∫ ((1 - 2sin^2(x)) / √(5 - 2u)) dx
= ∫ ((1 - 2sin^2(x)) / √(5 - 2u)) * (1 / (2cos(2x))) du
= (1/2) * ∫ ((1 - 2sin^2(x)) / (cos(2x) * √(5 - 2u))) du
= (1/2) * ∫ ((1 - 2sin^2(x)) / (1 - 2u)^(1/2)) du
Now you can integrate the expression with respect to u.
This is more a question to interpret the words:
" integral of the quotient of cosine of 2 times x and the square root of the quantity 5 minus 2 times the sine of 2 times x, dx"
which turns out to be:
∫cos(2x)dx/√(5-2sin(2x))
Check each case out by substituting u and du in the expression, and see which one seems to simplify most, or works best.