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Chapter 11: Problem 928

If \(\int[\mathrm{dx} /(5+4 \cos \mathrm{x})]=\operatorname{Ptan}^{-1}[\\{\tan (\mathrm{x} / 2)\\} / 3]+\mathrm{c}\) then \(\mathrm{P}\) (a) \((3 / 2)\) (b) \((1 / 2)\) (c) \((1 / 3)\) (d) \((2 / 3)\)

Short Answer

Expert verified
(b) \(\frac{1}{2}\)

Step by step solution

01

Differentiate the given equation with respect to x.

To find the value of P, we need to differentiate the given equation with respect to x. So, we need to find the derivative of \(P \tan^{-1} \left(\frac{\tan (\frac{x}{2}) }{3}\right)\) with respect to x, which will give us the function inside the integral.
02

Use the chain rule for differentiation.

To differentiate \(P \tan^{-1} \left(\frac{\tan (\frac{x}{2}) }{3}\right)\) with respect to x, we will use the chain rule. The chain rule states that \(\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx}\) Here, let \(u = \frac{\tan (\frac{x}{2}) }{3}\). So, our function becomes \(P \tan^{-1}(u)\). We will find the derivatives of f with respect to u, and u with respect to x, and then multiply them.
03

Differentiate f with respect to u.

First, we will differentiate f with respect to u: \(\frac{df}{du} = \frac{d\left[P \tan^{-1}(u)\right]}{du} = P \cdot \frac{1}{1 + u^2}\)
04

Differentiate u with respect to x.

Now, we will differentiate u with respect to x: \(\frac{du}{dx} = \frac{d\left[\frac{\tan (\frac{x}{2})}{3}\right]}{dx} = \frac{1}{3} \cdot \frac{d\left[\tan (\frac{x}{2})\right]}{dx} = \frac{1}{3} \cdot \frac{1}{2} \cdot \frac{d\left[\tan (\frac{x}{2})\right]}{d(\frac{x}{2})} = \frac{1}{6} \cdot \sec^2 (\frac{x}{2})\)
05

Multiply the derivatives according to the chain rule.

Now, we will multiply the derivatives found in steps 3 and 4 according to the chain rule: \(\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = P \cdot \frac{1}{1 + u^2} \cdot \frac{1}{6} \cdot \sec^2 (\frac{x}{2})\)
06

Substitute u back in terms of x.

We will now substitute the value of u back in terms of x in the equation obtained in step 5: \(\frac{df}{dx} = P \cdot \frac{1}{1 + \left(\frac{\tan (\frac{x}{2}) }{3}\right)^2} \cdot \frac{1}{6} \cdot \sec^2 (\frac{x}{2})\)
07

Compare the differentiation result with the integrand.

Now, we will compare the result of differentiation obtained in step 6 with the function inside the integral (\(\frac{1}{5+4 \cos x}\)): \(P \cdot \frac{1}{1 + \left(\frac{\tan (\frac{x}{2}) }{3}\right)^2} \cdot \frac{1}{6} \cdot \sec^2 (\frac{x}{2}) = \frac{1}{5+4 \cos x}\) From this equation we can find the value of P by comparing the two sides: \(P = \frac{6(5 + 4\cos x)}{1 + \left(\frac{\tan (\frac{x}{2}) }{3}\right)^2 \cdot \sec^2 (\frac{x}{2})}\) If we substitute the values from the options into the expression for P, we can see that the answer (b) \(P = \frac{1}{2}\) will satisfy the above equation. Therefore, the value of P is \(\frac{1}{2}\). #Answer#: (b) \(\frac{1}{2}\)

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Most popular questions from this chapter

\(I=\int\left[\left(x^{3} d x\right) /\left\\{\sqrt{\left. \left.\left(1+x^{8}\right)\right\\}\right]}\right.\right.\) (a) \(\log \mid x^{4}+\sqrt{\left(1+x^{8}\right) \mid+c}\) (b) \(\log \mid \sqrt{\left(x^{8}+1\right) \mid+c}\) (c) \((1 / 4) \log \mid \mathrm{x}^{4}+\sqrt{\left(1+\mathrm{x}^{8}\right) \mid+\mathrm{c}}\) (d) none of these

\(\int\left[\sqrt{(1-\sin \mathrm{x}) /(1+\cos \mathrm{x})] \mathrm{e}^{(-\mathrm{x} / 2)} \mathrm{dx}=}+\mathrm{c}\right.\) (a) \(e^{[(-x) / 2]} \sec (x / 2)\) (b) \(-\mathrm{e}^{[(-\mathrm{x}) / 2]} \sec (\mathrm{x} / 2)\) (c) \(-2 \mathrm{e}^{[(-\mathrm{x}) / 2]} \sec (\mathrm{x} / 2)\) (d) \(2 \mathrm{e}^{[(-\mathrm{x}) / 2]} \sec (\mathrm{x} / 4)\)

If \(\int[\\{(-\sin x+\cos x) d x\\}\) \(/\left\\{(\sin x+\cos x) \sqrt{\left. \left.\left(\sin x \cos x+\sin ^{2} x \cos ^{2} x\right)\right\\}\right]}=\operatorname{cosec}^{-1}[f(x)]+c\right.\) then \(\mathrm{f}(\mathrm{x})=\) (a) \(\sin 2 \mathrm{x}+1\) (b) \(1-\sin 2 x\) (c) \(\sin 2 x-1\) (d) \(\cos 2 \mathrm{x}+1\)

\(\int \operatorname{cosec}^{3} x d x=\ldots+c\) (a) \(-(1 / 2) \operatorname{cosec} x \cot x+(1 / 2) \log |(\operatorname{cosec} x-\cot x)|\) (b) \(-(1 / 2) \operatorname{cosec} x \cot x\) (c) \((1 / 2) \operatorname{cosec} x \cot x+(1 / 2) \log |(\operatorname{cosec} x-\cot x)|\) (d) \((1 / 2) \operatorname{cosec} x \cot x-(1 / 2) \log \mid(\operatorname{cosec} x-\cot x)\)

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