Integral of cosec^2(x)sec^2(x) (substitution)
Integral of cosec^2(x)sec^2(x) (substitution)
Problem:
∫ csc²(x) sec²(x) dx
Step 1: Use Substitution
We start by using the identity \( \sec²(x) = 1 + \tan²(x) \), but it’s easier to use a substitution directly.
Let \( u = \tan(x) \), so that \( du = \sec²(x) dx \).
This transforms the integral into:
∫ csc²(x) * du
Step 2: Express csc²(x) in Terms of u
We know that \( \csc²(x) = 1 + \cot²(x) \), and \( \cot(x) = 1 / u \) since \( \tan(x) = u \). Therefore, \( \cot²(x) = 1/u² \).
Thus, \( \csc²(x) = 1 + (1/u²) \).
Step 3: Substitute and Simplify
Now, substitute this back into the integral:
∫ \left( 1 + \frac{1}{u²} \right) du
Step 4: Integrate the Resulting Expression
Now, integrate each term:
∫ 1 du = u
∫ \frac{1}{u²} du = -\frac{1}{u}
Step 5: Substitute Back for u
Since \( u = \tan(x) \), substitute back:
∫ csc²(x) sec²(x) dx = tan(x) - \frac{1}{\tan(x)} + C
Final Answer:
∫ csc²(x) sec²(x) dx = tan(x) - cot(x) + C