Find the indefinite integral. integral csc^2(x/5)dx Observe that is the derivative of 1/5x. Let g(x) = 1/5x. Multiply and divide the given integral integral csc^2(x/5) dx by 1/5. Therefore, the integral becomes integral csc^2 (x/5)(1/5) dx. Now, g(x) = x/5, g'(x) = 1/5. Define f(g(x)) = csc^2(x/5), such that f(x) = csc^2 Rewrite the given integral in terms of g(x), where F(g(x)) is the antiderivative of f(g(x)). integral csc^2 (x/5) dx = 5 integral csc^2 (x/5)(1/5)dx = 5 integral f((x))g'(x) dx = 5
The Correct Answer and Explanation is :
To find the indefinite integral of ( \int \csc^2\left(\frac{x}{5}\right) \, dx ), we will use substitution and the known properties of antiderivatives.
Step 1: Set up the substitution
Notice that the integral involves ( \csc^2\left(\frac{x}{5}\right) ), which is a function of ( x ). We will simplify the integral using a substitution. Let:
[
g(x) = \frac{x}{5}
]
Therefore, the derivative of ( g(x) ) is:
[
g'(x) = \frac{1}{5}
]
This substitution will help simplify the integral. Now, we can rewrite the integral in terms of ( g(x) ).
Step 2: Modify the integral
To apply the substitution correctly, we need to account for the factor of ( \frac{1}{5} ). Thus, multiply and divide the given integral by ( \frac{1}{5} ) to adjust for the substitution:
[
\int \csc^2\left(\frac{x}{5}\right) \, dx = 5 \int \csc^2\left(g(x)\right) g'(x) \, dx
]
Now, the integral has a familiar form, where ( f(x) = \csc^2(x) ) and the antiderivative of ( \csc^2(x) ) is ( -\cot(x) ). Therefore:
[
\int \csc^2(x) \, dx = -\cot(x)
]
Step 3: Apply the antiderivative
We now apply the antiderivative to the integral in terms of ( g(x) ):
[
5 \int \csc^2\left(g(x)\right) g'(x) \, dx = 5 \left[ -\cot\left(g(x)\right) \right]
]
Substitute ( g(x) = \frac{x}{5} ) back into the equation:
[
= 5 \left[ -\cot\left(\frac{x}{5}\right) \right] + C
]
Step 4: Final answer
The final result is:
[
\int \csc^2\left(\frac{x}{5}\right) \, dx = -5 \cot\left(\frac{x}{5}\right) + C
]
Explanation:
In this solution, we used substitution to simplify the integral. By letting ( g(x) = \frac{x}{5} ), we made the integrand easier to handle, turning it into a familiar form involving ( \csc^2(x) ), whose antiderivative is well-known. This approach utilizes the chain rule, ensuring we multiply by the correct constant to account for the substitution.
In summary, substitution is a powerful technique that allows us to transform more complicated integrals into simpler ones by using known antiderivatives. In this case, recognizing the derivative relationship between ( \csc^2(x) ) and ( \cot(x) ) was crucial to finding the solution.