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How do you find the integral of # [ (1+cos^2(x)) / sin(x) ] dx#?

Answer 1

Isabella Ackerman

#-2ln|cscx +cotx| +cos x +C#

Rewrite the integrand as #[(1+1-sin^2 x)/sinx]= [(2-sin^2 x)/sinx]=2cscx -sinx#

Now #int csc x dx= -ln|csc x+cotx| # and #int sinx dx= -cos x#

The required Integral would be #-2ln|cscx +cotx| +cos x +C#

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Answer 2

Emilia Aguilar

To find the integral of [(1 + cos^2(x)) / sin(x)] dx, you can use the trigonometric identity cos^2(x) = 1 - sin^2(x) to simplify the expression. After simplification, the integral can be written as ∫ (1 + 1 - sin^2(x)) / sin(x) dx. Then, you can split this expression into two separate integrals: ∫ (1/sin(x)) dx + ∫ (1 - sin^2(x)) / sin(x) dx. The first integral can be evaluated as -ln|csc(x) + cot(x)| + C, where C is the constant of integration. The second integral can be simplified further using substitution or trigonometric identities.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.