Math Problem Statement

Find ∫\frac{\sin ^{2} x-\cos ^{2} x}{\sin x \cos x} dx

Solution

To solve the integral

I=sin2xcos2xsinxcosxdx,I = \int \frac{\sin^2 x - \cos^2 x}{\sin x \cos x} \, dx,

we start by simplifying the integrand using trigonometric identities.

Step 1: Simplify sin2xcos2x\sin^2 x - \cos^2 x

Recall the identity: sin2xcos2x=cos(2x).\sin^2 x - \cos^2 x = -\cos(2x).

Thus, the integrand becomes: sin2xcos2xsinxcosx=cos(2x)sinxcosx.\frac{\sin^2 x - \cos^2 x}{\sin x \cos x} = \frac{-\cos(2x)}{\sin x \cos x}.

Step 2: Simplify further

Using the double-angle identity for sine, sin(2x)=2sinxcosx\sin(2x) = 2\sin x \cos x, we can rewrite: cos(2x)sinxcosx=cos(2x)12sin(2x)=2cos(2x)sin(2x).\frac{-\cos(2x)}{\sin x \cos x} = -\frac{\cos(2x)}{\frac{1}{2} \sin(2x)} = -2 \cdot \frac{\cos(2x)}{\sin(2x)}.

This simplifies to: sin2xcos2xsinxcosx=2cot(2x).\frac{\sin^2 x - \cos^2 x}{\sin x \cos x} = -2 \cot(2x).

Step 3: Integrate

The integral now becomes: I=2cot(2x)dx.I = \int -2 \cot(2x) \, dx.

The integral of cot(2x)\cot(2x) is: cot(2x)dx=12lnsin(2x)+C.\int \cot(2x) \, dx = \frac{1}{2} \ln|\sin(2x)| + C.

Thus: I=212lnsin(2x)+C=lnsin(2x)+C.I = -2 \cdot \frac{1}{2} \ln|\sin(2x)| + C = -\ln|\sin(2x)| + C.

Final Answer:

sin2xcos2xsinxcosxdx=lnsin(2x)+C.\int \frac{\sin^2 x - \cos^2 x}{\sin x \cos x} \, dx = -\ln|\sin(2x)| + C.

Would you like a detailed explanation of any step? Here are some related questions:

  1. What are the key trigonometric identities used in simplifying integrals?
  2. How does the double-angle formula for sine simplify such expressions?
  3. Why is cot(2x)\cot(2x) related to lnsin(2x)\ln|\sin(2x)| in its integral?
  4. Can this integral be approached using substitution?
  5. How would the integral change if sin2x+cos2x\sin^2 x + \cos^2 x were in the numerator instead?

Tip: Memorizing the derivatives and integrals of basic trigonometric functions like sinx,cosx,tanx,\sin x, \cos x, \tan x, and cotx\cot x 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