5 days ago · The log sine function, also called the logsine function, is defined by S_n=int_0^pi [ln (sinx)]^ndx. (1) The first few cases are given by S_1 = -piln2 (2) S_2 = 1/ (12)pi^3+pi (ln2)^2 …

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Sine is the unique solution with y(0) = 0 and y′(0) = 1; cosine is the unique solution with y(0) = 1 and y′(0) = 0. One can then prove, as a theorem, that solutions , are periodic, having the same …

In symbols, log a6 = log a+log b. 2. The logarithm of a fraction is equal to the logarithm. of the numerator minus the logarithm of the denominator. In symbols, log T log a — = log h. 3. The …

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Natural trigonometric functions are expressed as. Trigonometric functions ranging 0 to 90 degrees are tabulated below: arcsin (a) = sin-1(a) (1a) arccos (a) = cos-1(a) (2a) arctan (a) = tan-1(a) (3a)

Dec 16, 2024 · Example 34 Evaluate ∫_0^(𝜋/2 ) log⁡sin⁡𝑥 𝑑𝑥 Let I1=∫_0^(𝜋/2 ) 𝑙𝑜𝑔(𝑠𝑖𝑛𝑥) 𝑑𝑥 ∴ I1=∫_0^(𝜋/2) 𝑠𝑖𝑛(𝜋/2−𝑥)𝑑𝑥 I1= ∫_0^(𝜋/2) 𝑙𝑜𝑔(cos⁡𝑥 )𝑑𝑥 Adding (1) and (2) i.e. (1) + (2) I1+ I1=∫_0^(𝜋/2) 〖𝑙𝑜𝑔(sin⁡𝑥 )𝑑𝑥+∫_0^(𝜋/2 ...

If you are familiar with AM-GM inequality, here is a cute solution: Let $a=\log _{\sin(x)\cos(x)}(\sin(x)), b= \log_{\sin(x)\cos(x)}(\cos(x))$. You know $$a+b=1 \\ …

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