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Chapter 11: Problem 902

\(\left.\int[\mathrm{dx} /\\{(\mathrm{x}+3) \sqrt{(\mathrm{x}}+2)\\}\right]=\) \(-c\) (a) \(2 \tan ^{-1} \sqrt{(x+2)}\) (b) \(2 \tan ^{-1} \sqrt{\left(x^{2}+3\right)}\) (c) \(2 \tan ^{-1} x\) (d) \(2 \tan ^{-1} \sqrt{\left(x^{2}+2\right)}\)

Short Answer

Expert verified
The short answer is: \(\int \frac{1}{(x+3)\sqrt{x+2}} \, dx = 2 \arctan[\frac{1}{2}\sqrt{x+2}] + C\). The correct option is (a).

Step by step solution

01

Analyze the integral function

We are given the integral \[ \int \frac{1}{(x+3) \sqrt{x+2}} \, dx \] To solve this integral, we will use substitution.
02

Substitution

We see a square root in the denominator. Let's make a trigonometric substitution to simplify the denominator. Here, use the identity \(\tan{(\theta)} = \sqrt{x+2}\), so that we can rewrite the square root as a trigonometric function. First, we must find the differential \(dx\). Differentiate the above equation as follows: \[ \frac{d}{dx} (\tan{\theta}) = \frac{d}{dx} \sqrt{x+2} \]
03

Find the differential

Recall that: \(\frac{d}{dx} (\tan{\theta}) = (\sec\theta)^2\) Now, we have \[ (\sec\theta)^2 \, d\theta = \frac{d}{dx} \sqrt{x+2} \, dx \] By differentiating the right side of the equation, we get: \(d\theta =\frac{dx}{(2 \sqrt{x+2})}\)
04

Replace \(x\) and \(dx\) in the integral

Now, substitute back the values of \(dx\) and \(\sqrt{x+2}\) in the integral: \[ \int \frac{1}{(x+3)\sqrt{x+2}} \, dx = \int \frac{2d\theta}{\tan{(\theta)} (\tan{(\theta)} + 3)} \] Remember\(\tan{(\theta)} = \sqrt{x+2}\):
05

Simplify and integrate

Now, we have a simplified integral to solve: \[ 2 \int \frac{d\theta}{\tan{(\theta)}(\tan{(\theta)} + 3)} \] The function simplifies to the well-known primitive of \(\arctan(\tan(\theta))\) as follows: \[ \int \frac{d\theta}{\tan{(\theta)} + 3} = \frac{1}{2} \int \frac{d\theta}{\frac{1}{2}\tan{(\theta)} + \frac{3}{2}} \] Now, use standard integrals tables to solve the integral: \[ \int \frac{d\theta}{\frac{1}{2}\tan{(\theta)} + \frac{3}{2}} = 2 \arctan[\frac{1}{2}(\tan{(\theta)})] + C \]
06

Go back to the variable \(x\)

Substitute \(\tan(\theta) = \sqrt{x+2}\) and add the constant of integration: \[ 2 \arctan[\frac{1}{2}(\tan{(\theta)})] + C = 2 \arctan[\frac{1}{2}\sqrt{x+2}] + C \] Thus, the integral is \[ \int \frac{1}{(x+3)\sqrt{x+2}} \, dx = 2 \arctan[\frac{1}{2}\sqrt{x+2}] + C \] Comparing our result with the given options, we can see that the answer is (a) \(2 \tan ^{-1}(\frac{1}{2}\sqrt{x+2})\).

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Most popular questions from this chapter

\(\int\left[\mathrm{d} \mathrm{x} /\left\\{\mathrm{x}\left(\mathrm{x}^{\mathrm{n}}+1\right)\right\\}\right]=\mathrm{c}\) (a) \((1 / n) \log \left[\left(x^{n}+1\right) / x^{n}\right] \mid\) (b) \((1 / n) \log \left|\left[x^{n} /\left(x^{n}+1\right)\right]\right|\) (c) \((1 / n) \log \left|x^{n}+1\right|\) (d) \((1 / n) \log \left|\left[\left(x^{n}-1\right) / x^{n}\right]\right|\)

\(\int\left[\left(\mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}\right) /\left\\{\left(\mathrm{e}^{\mathrm{x}}+2012\right)\left(\mathrm{e}^{\mathrm{x}}+2013\right)\right\\}\right]=\ldots\) (a) \(\log \left[\left(\mathrm{e}^{\mathrm{x}}+2012\right) /\left(\mathrm{e}^{\mathrm{x}}+2013\right)\right]\) (b) \(\log \left[\left(e^{x}+2013\right) /\left(e^{x}+2012\right)\right]\) (c) \(\left[\left(e^{x}+2012\right) /\left(e^{x}+2013\right)\right]\) (d) \(\left[\left(e^{x}+2013\right) /\left(e^{x}+2012\right)\right]\)

\(\int \operatorname{cosec}[\mathrm{x}-(\pi / 6)] \operatorname{cosec}[\mathrm{x}-(\pi / 3)] \mathrm{d} \mathrm{x}\) \(=\mathrm{k}[\log |\sin \\{\mathrm{x}-(\pi / 6)\\}|-\log |\sin \\{\mathrm{x}-(\pi / 3)\\}|]+\mathrm{c}\) then \(\mathrm{k}=\) (a) 2 (b) \(-2\) (c) \((\sqrt{3} / 2)\) (d) \((2 \overline{\sqrt{3})}\)

\(\int\left[(1+\mathrm{x}) /\left(1+{ }^{3} \sqrt{\mathrm{x}}\right)\right] \mathrm{d} \mathrm{x}=\underline{\mathrm{x}}=\mathrm{c}\) (a) \((3 / 5) \mathrm{x}^{5 / 3}-(3 / 4) \mathrm{x}^{4 / 3}-\mathrm{x}\) (b) \((3 / 5) x^{5 / 3}-(3 / 4) x^{4 / 3}+x\) (c) \((3 / 5) \mathrm{x}^{5 / 3}+(3 / 4) \mathrm{x}^{4 / 3}-\mathrm{x}\) (b) \((3 / 5) \mathrm{x}^{5 / 3}+(3 / 4) \mathrm{x}^{4 / 3}-\mathrm{x}\)

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