What is the integral of #int (cos(x)/sin(x)-1)dx#?
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The integral of (\int (\frac{\cos(x)}{\sin(x)} - 1) dx) is (-\ln|\sin(x)| - x + C), where (C) is the constant of integration.
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The integral of ( \int \left(\frac{\cos(x)}{\sin(x)} - 1\right)dx ) can be solved as follows:
- Rewrite the expression as ( \int \left(\frac{\cos(x)}{\sin(x)} - \frac{\sin(x)}{\sin(x)}\right)dx ).
- Combine the terms under the same fraction: ( \int \frac{\cos(x) - \sin(x)}{\sin(x)}dx ).
- Use the trigonometric identity ( \cos(x) - \sin(x) = -\sqrt{2}\sin\left(x + \frac{\pi}{4}\right) ).
- So, the integral becomes ( -\sqrt{2} \int \frac{\sin\left(x + \frac{\pi}{4}\right)}{\sin(x)}dx ).
- This integral can be solved by u-substitution: Let ( u = x + \frac{\pi}{4} ), then ( du = dx ).
- Substituting back, we get ( -\sqrt{2} \int \frac{\sin(u)}{\sin(u - \frac{\pi}{4})}du ).
- Using the identity ( \frac{\sin(u)}{\sin(u - \frac{\pi}{4})} = \frac{\sin(u)}{\sin(u)\cos(\frac{\pi}{4}) + \cos(u)\sin(\frac{\pi}{4})} = \frac{\sin(u)}{\frac{\sqrt{2}}{2}\sin(u) + \frac{\sqrt{2}}{2}\cos(u)} ).
- After simplification, the integral becomes ( -2\sqrt{2} \int \frac{\sin(u)}{\sin(u) + \cos(u)}du ).
- This integral can be solved using another u-substitution: Let ( v = \tan\left(\frac{u}{2}\right) ), then ( dv = \frac{1}{2}\sec^2\left(\frac{u}{2}\right)du ).
- Substituting back, we get ( -4\sqrt{2} \int \frac{dv}{1 + v^2} ).
- This integral represents the arctangent function: ( -4\sqrt{2} \arctan(v) + C ).
- Substituting back ( v = \tan\left(\frac{u}{2}\right) ) and ( u = x + \frac{\pi}{4} ), the final result is ( \boxed{-4\sqrt{2} \arctan\left(\tan\left(\frac{x + \frac{\pi}{4}}{2}\right)\right) + C} ).
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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.
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.
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.
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.
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