# Integration ∫ [√(sin^2 x-3sin x+2))/√(sin^2 x+3sin x+2))]dx

• MHB
• juantheron
In summary, the given integral can be evaluated by using trigonometric substitutions and then applying the logarithmic and inverse tangent functions to get the final solution.
juantheron
Evaluation of $\displaystyle \int \sqrt{\frac{\sin^2 x-3\sin x+2}{\sin^2 x+3\sin x+2}}dx$

Solution [sp]Let $\displaystyle I = \int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx$

We can write $\displaystyle \sqrt{\frac{1-\sin x}{1+\sin x}} = \sqrt{\frac{1-\sin x}{1+\sin x}\times \frac{1+\sin x}{1+\sin x}} = \frac{\cos x}{1+\sin x}$So we get $\displaystyle I = \int\frac{\cos x}{1+\sin x}\cdot \sqrt{\frac{2-\sin x}{2+\sin x}}dx$Now Let $1+\sin x= y\;,$ Then $\cos xdx = dy$So Integral $\displaystyle I = \int\frac{1}{y}\cdot \sqrt{\frac{3-y}{1+y}}dy$Now Put $\displaystyle \frac{3-y}{1+y}=t^2\Rightarrow y=\frac{3-t^2}{1+t^2}$So we get $\displaystyle y=-\left[1-\frac{4}{1+t^2}\right] = \left[\frac{4}{1+t^2}-1\right].$

So $\displaystyle dy = -\frac{8t}{(1+t^2)^2}$So Integral $\displaystyle I = \int\frac{1+t^2}{3-t^2}\cdot t\cdot \frac{-8t}{(1+t^2)^2}dt = 8\int\frac{t^2}{(t^2-3)\cdot (1+t^2)}dt$So Integral $\displaystyle I = 2\int \left[\frac{3(t^2+1)+(t^2-3)}{(t^2-3)\cdot (1+t^2)}\right]dt = 2\int \left[\frac{3}{t^2-(\sqrt{3})^2}+\frac{1}{1+t^2}\right]dt$So Integral $\displaystyle I = 6\cdot \frac{1}{2\sqrt{3}}\cdot \ln\left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+2\tan^{-1}(t)+\mathcal{C}$So Integral $\displaystyle I = \sqrt{3}\cdot \ln\left|\frac{\sqrt{2-\sin x}-\sqrt{3}\cdot \sqrt{2+\sin x}}{\sqrt{2-\sin x}-\sqrt{3}\cdot \sqrt{2+\sin x}}\right|+2\tan^{-1}\left(\sqrt{\frac{2-\sin x}{2+\sin x}}\right)+\mathcal{C}$[/sp]

## 1. What is integration?

Integration is a mathematical process of finding the area under a curve or the accumulation of a quantity over a given interval. It is the inverse operation of differentiation.

## 2. What is the purpose of integration?

The purpose of integration is to solve problems involving the accumulation of quantities, such as finding the distance traveled by an object, the volume of a shape, or the work done by a force.

## 3. How do you solve the integral of √(sin^2 x-3sin x+2)/√(sin^2 x+3sin x+2) dx?

To solve this integral, you can use the substitution method by letting u = sin x. This will simplify the integral to ∫ [√(u^2-3u+2)/√(u^2+3u+2)]du. Then, you can factor the numerator and denominator and use the trigonometric identity sin^2 x + cos^2 x = 1 to simplify the integral further.

## 4. What are the steps for solving an integral using the substitution method?

The steps for solving an integral using the substitution method are:

1. Identify a suitable substitution (usually a function within the integral)
2. Apply the substitution and rewrite the integral in terms of the new variable
3. Simplify the integral using algebraic or trigonometric identities
4. Integrate the simplified integral
5. Substitute the original variable back in the final answer

## 5. Can you solve the integral of √(sin^2 x-3sin x+2)/√(sin^2 x+3sin x+2) dx without using the substitution method?

Yes, you can also solve this integral by using the partial fraction decomposition method. This involves breaking the fraction into smaller fractions with simpler denominators and then integrating each term separately. However, the substitution method is often simpler and more efficient for this particular integral.

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