Question

Find the integration of
$\displaystyle \sf \: \int \frac{ \sin^{2}x - \cos ^{2} x}{ \sin \: x. \cos \: x} \: \: dx$
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Answer 1

★ Formula Applied :

$\sf \bullet\ \; cos^2x-sin^2x=cos2x\\\\\to\ \sf \pink{sin^2x-cos^2x=-cos2x}$

$\bullet\ \; \sf sin2x=2.sinx.cosx\\\\\to \sf \blue{sinx.cosx=\dfrac{sin2x}{2}}$

$\displaystyle \bullet\ \; \sf \int \dfrac{dx}{x}=ln|x|$

$\bullet\ \; \sf \ln (ab)=\ln a+\ln b$

★ Explanation :

$\displaystyle \sf \int \dfrac{sin^2x-cos^2x}{sinx.cosx}dx\\\\\to \sf \int \dfrac{-cos2x}{\frac{sin2x}{2}}dx\\\\\to \sf \int \dfrac{-2cos2x}{sin2x}dx$

Lets use substitution method ,

Let , u = sin2x

⇒ du = 2.cos2x.dx

$\to \displaystyle \sf \int \dfrac{-du}{u}\\\\\to \sf -\int \dfrac{du}{u}\\\\\to \sf -ln|u|$

$\to \sf \red{-ln|sin2x|+c}$

$\to \sf - \ln |2sinx.cosx|+c$

$\to \sf - \ln |sinx.cosx|+(\ln 2+c)$

$\leadsto - \sf \red{\ln |sinx.cosx|+c}\ \; \bigstar$

★ Alternate Method :

$\displaystyle \sf \int \dfrac{sin^2x-cos^2x}{sinx.cosx}dx$

$\displaystyle \to \sf \int \left( \dfrac{sin^2x}{sinx.cosx}-\dfrac{cos^2x}{sinx.cosx}\right)dx$

$\displaystyle \to \sf \int \left( \dfrac{sinx}{cosx} -\dfrac{cosx}{sinx} \right)dx\\\\\to\ \sf \int (tanx-cotx)dx\\\\\to \sf \int tanx.dx-\int cotx.dx\\\\\to \sf ln|secx|-ln|sinx|+c$

$\to \sf ln\left| \dfrac{secx}{sinx}\right|+c$

$\to \sf ln\left| \dfrac{1}{sinx.cosx} \right|+c$

$\leadsto \sf \pink{-ln|sinx.cosx|+c}\ \; \bigstar$

Answer 2

Answer :-

• I = -log | sinx.cosx | + c

↪[ Refer to the attachment ]

Learn More :-

↪Integration Definition :-

• The integration denotes the summation of discrete data. The integral is calculated to find the functions which will describe the area, displacement, volume, that occurs due to a collection of small data, which cannot be measured singularly. In a broad sense, in calculus, the idea of limit is used where algebra and geometry are implemented. Limits help us in the study of the result of points on a graph such as how they get closer to each other until their distance is almost zero. We know that there are two major types of calculus :-

• Differential Calculus
• Integral Calculus

Example :-

Question :-

• Find the integral of the function :

∫x² dx.

Solution :-

↪ ∫x² dx

↪ (x³/3) + C.