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Natali5045456 [20]
3 years ago
5

Give me your pardon, sir: I have done you wrong: But pardon't, as you are a gentleman. This presence knows, and you must needs h

ave heard, How I am punish'd with sore distraction. What I have done That might your nature, honour, and exception Roughly awake, I here proclaim was madness. Was't Hamlet wrong'd Laertes? Never Hamlet: If Hamlet from himself be ta'en away, And when he's not himself does wrong Laertes, Then Hamlet does it not, Hamlet denies it. Who does it, then? His madness: if't be so, Hamlet is of the faction that is wrong'd; His madness is poor Hamlet's enemy. Sir, in this audience, Let my disclaiming from a purpos'd evil Free me so far in your most generous thoughts That I have shot my arrow o'er the house And hurt my brother.
English
1 answer:
erastovalidia [21]3 years ago
8 0

What does the question ask?

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★ Formula Applied :

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

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\displaystyle \bullet\ \; \sf \int \dfrac{dx}{x}

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★ Explanation :

\begin{gathered}\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\end{gathered}

Lets use substitution method ,

Let , u = sin2x

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\begin{gathered}\to \displaystyle \sf \int \dfrac{-du}{u}\\\\\to \sf -\int \dfrac{du}{u}\\\\\to \sf -ln|u|\\\\\end{gathered}

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★ Alternate Method :

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

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\begin{gathered}\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\end{gathered}

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

\begin{gathered}\to \sf ln\left| \dfrac{1}{sinx.cosx} \right|+c\\\\\end{gathered}]

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

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