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juantheron
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Evaluation of $\displaystyle \int \sqrt{\frac{\sin^2 x-3\sin x+2}{\sin^2 x+3\sin x+2}}dx$
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.
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.
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.
The steps for solving an integral using the substitution method are:
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.