Math Problem Statement
Find ∫\frac{\sin ^{2} x-\cos ^{2} x}{\sin x \cos x} dx
Solution
To solve the integral
we start by simplifying the integrand using trigonometric identities.
Step 1: Simplify
Recall the identity:
Thus, the integrand becomes:
Step 2: Simplify further
Using the double-angle identity for sine, , we can rewrite:
This simplifies to:
Step 3: Integrate
The integral now becomes:
The integral of is:
Thus:
Final Answer:
Would you like a detailed explanation of any step? Here are some related questions:
- What are the key trigonometric identities used in simplifying integrals?
- How does the double-angle formula for sine simplify such expressions?
- Why is related to in its integral?
- Can this integral be approached using substitution?
- How would the integral change if were in the numerator instead?
Tip: Memorizing the derivatives and integrals of basic trigonometric functions like and can greatly simplify solving such integrals.
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Math Problem Analysis
Mathematical Concepts
Trigonometric Identities
Integration
Simplification of Rational Functions
Formulas
sin^2 x - cos^2 x = -cos(2x)
sin(2x) = 2sin(x)cos(x)
Integral of cot(kx): ∫cot(kx) dx = (1/k)ln|sin(kx)| + C
Theorems
Double-Angle Formulas
Logarithmic Integration Rule
Suitable Grade Level
Grades 11-12 or College Level
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