Step-by-Step Guide | Evaluating the Integral of csc(u) du Using Trigonometric Substitution

∫ cscu du

To evaluate the integral ∫ csc(u) du, where csc(u) represents the cosecant function of u, we can use trigonometric identities and make a substitution

To evaluate the integral ∫ csc(u) du, where csc(u) represents the cosecant function of u, we can use trigonometric identities and make a substitution.

Step 1: Rewriting the integral using trigonometric identities
The cosecant function can be expressed in terms of sine:
csc(u) = 1/sin(u)

So, we can rewrite the integral as:
∫ (1/sin(u)) du

Step 2: Making a substitution
Let’s make the substitution:
x = cos(u)

Taking the derivative of x with respect to u, we have:
dx/dx = -sin(u)
du = -dx/sin(u)
du = -dx/√(1 – x^2) (using the Pythagorean identity sin^2(u) + cos^2(u) = 1)

Step 3: Substituting into the integral
Now, let’s substitute these values into the integral:
∫ (1/sin(u)) du = ∫ (1/sin(u)) (-dx/√(1 – x^2))

Since csc(u) = 1/sin(u), our integral becomes:
– ∫ (1/√(1 – x^2)) dx

Step 4: Evaluating the integral
The integral ∫ (1/√(1 – x^2)) dx corresponds to the arc sine function, or sin^(-1)(x). Therefore, we have:
∫ (1/sin(u)) du = -sin^(-1)(x) + C

Step 5: Converting back to the original variable
We need to express our answer in terms of the original variable u, so let’s substitute x = cos(u) back into the result:
∫ csc(u) du = -sin^(-1)(cos(u)) + C

So, the final answer to the integral is:
∫ csc(u) du = -sin^(-1)(cos(u)) + C, where C represents the constant of integration.

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