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Chapter 11: Problem 985

\(\mathrm{I}=\int \cot ^{2} \mathrm{x} \cdot \operatorname{cosec}^{4} \mathrm{xdx}=-\left(\mathrm{k}^{3} / 3\right)-\left(\mathrm{k}^{5} / 5\right)+\mathrm{c}\) then \(\mathrm{k}\) is (a) \(\sin x \cdot \cos x\) (b) \(\cot x\) (c) \(-\cot x\) (d) \(\tan \mathrm{x}\)

Short Answer

Expert verified
The short answer to the integration question is (c) \(-\cot(x)\).

Step by step solution

01

Identify relevant trigonometric identities and substitution

We need to identify relevant trigonometric identities and substitution to solve the integration. We know that \(\cot(x) = \frac{1}{\tan(x)}\) and \(\cosec(x) = \frac{1}{\sin(x)}\). Using these relationships, we can rewrite our integrand function: \(\cot^2(x) \cdot \cosec^4(x) = \frac{1}{\tan^2(x)\sin^4(x)}\).
02

Express the integrand function in terms of sin(x) and cos(x) functions

We can now express the integrand function in terms of\(\sin(x)\) and \(\cos(x)\) functions: \(\cot^2(x) \cdot \cosec^4(x) = \frac{1}{\tan^2(x)\sin^4(x)} = \frac{\cos^2(x)}{\sin^2(x) \cdot \sin^4(x)} = \frac{\cos^2(x)}{\sin^6(x)}\).
03

Apply substitution for integration

Let's apply the substitution: \(u = \sin(x)\), so \(du=\cos(x)dx\). Using this substitution, our integral becomes: \(\mathrm{I} = \int \frac{\cos^2(x) dx}{\sin^6(x)} = \int \frac{du}{u^6}\).
04

Integrate the simplified function

Now, we'll integrate the function with respect to u: \(\mathrm{I} = \int \frac{du}{u^6} = -\frac{1}{5u^5} + C\). At this step, we have found the integration of the simplified function.
05

Replace u with the original function sin(x)

Now, we'll replace u back with the original function \(\sin(x)\): \(\mathrm{I} = -\frac{1}{5\sin^5(x)} + C\).
06

Compare with the given expression and extract the value of k

Our final answer should have the form \(- \left(k^3 / 3\right) - \left(k^5 / 5\right) + c\). Comparing this expression with \(\mathrm{I} = -\frac{1}{5\sin^5(x)} + C\), we can observe that \(k = -\cot(x)\). Therefore, our correct answer is (c) \(-\cot(x)\).

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Most popular questions from this chapter

If \(\int\left[\left(3^{x}-1\right) /\left(3^{x}+1\right)\right] d x=k \log \left|3^{(x / 2)}+3^{-(x / 2)}\right|+c\) then \(k=\) (a) \(\log _{3} \mathrm{e}\) (b) \(\log _{\mathrm{e}} 3\) (c) \(2 \log _{3} \mathrm{e}\) (d) \(2 \log _{\mathrm{e}} 3\)

\(\int \mathrm{e}^{2 \mathrm{x}+\log \mathrm{x}} \mathrm{dx}=\) (a) \((1 / 4)(2 \mathrm{x}-1) \mathrm{e}^{2 \mathrm{x}}\) (b) \((1 / 2)(2 \mathrm{x}-1) \mathrm{e}^{2 \mathrm{x}}\) (c) \((1 / 4)(2 \mathrm{x}+1) \mathrm{e}^{2 \mathrm{x}}\) (d) \((1 / 4)(2 \mathrm{x}+1) \mathrm{e}^{2 \mathrm{x}}\)

\(\int \sqrt{(1+\operatorname{cosec} x d x)}=\ldots\) (a) \(2 \sin ^{-1}(\sqrt{\cos } \mathrm{x})\) (b) \(2 \cos ^{-1}(\sqrt{\sin } \mathrm{x})\) (c) \(\sin ^{-1}(2 \sin x-1)\) (d) \(2 \cos ^{-1}(\sqrt{\cos x})\)

If \(\int\left(x^{30}+x^{20}+x^{10}\right)\left(2 x^{20}+3 x^{10}+6\right)^{(1 / 10)} d x\) \(=\mathrm{k}\left(2 \mathrm{x}^{30}+3 \mathrm{x}^{20}+6 \mathrm{x}^{10}\right)^{(11 / 10)}+\mathrm{c}\) then \(\mathrm{k}=\) (c) \((1 / 66)\) (a) \((1 / 60)\) (b) \(-(1 / 60)\) (c) \(-(1 / 66)\)

\(\int e^{x}\left[\left(x^{3}-x-2\right) /\left(x^{2}+1\right)^{2}\right] d x=\) (a) \(e^{x}\left[(2 x-1) /\left(x^{2}+1\right)\right]\) (b) \(e^{x}\left[(x+1) /\left(x^{2}+1\right)\right]\) (c) \(e^{x}\left[(x-1) /\left(x^{2}+1\right)\right]\) (b) \(e^{x}\left[(2 x-2) /\left(x^{2}+1\right)\right]\)
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