Question

Dx/(sin^2 (2 x)*cos^2 (2x))

Answer 1

Ответ:

$\displaystyle \bf \int\limits {\frac{dx}{sin^2(2x)\;cos^2(2x)} }=\frac{1}{2}tg(2x)-\frac{1}{2}ctg(2x)+C$

Пошаговое объяснение:

Вычислить неопределенный интеграл:

$\displaystyle \bf \int\limits {\frac{dx}{sin^2(2x)\;cos^2(2x)} }$

Преобразуем подинтегральное выражение.

• Основное тригонометрическое тождество:

          sin²α + cos²α = 1

$\displaystyle \bf \int\limits {\frac{sin^2(2x)+cos^2(2x)}{sin^2(2x)\;cos^2(2x)} } \, dx =\\\\=\int\limits {\frac{sin^2(2x)}{sin^2(2x)\;cos^2(2x)} } \, dx+\int\limits {\frac{cos^2(2x)}{sin^2(2x)\;cos^2(2x)} } \, dx=\\\\=\int\limits {\frac{dx}{cos^2(2x)} } +\int\limits {\frac{dx}{sin^2(2x)} } =\\\\=\frac{1}{2} \int\limits {\frac{d(2x)}{cos^2(2x)} } +\frac{1}{2} \int\limits {\frac{d(2x)}{sin^2(2x)} } =$

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Получили табличные интегралы:

$\boxed {\displaystyle \bf \int\limits {\frac{dx}{sin^2x} } =-ctg\;x+C }\\\\\boxed {\displaystyle \bf \int\limits {\frac{dx}{cos^2x} } =tg\;x+C }$

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$\displaystyle \bf =\frac{1}{2}tg(2x)-\frac{1}{2}ctg(2x)+C$

Answer 2

To simplify the expression, let's work step by step:

Given expression: dx / (sin^2(2x) * cos^2(2x))

We can simplify this expression using trigonometric identities. The double-angle identities for sine and cosine will be useful here:

sin^2(2x) = (1 - cos(4x)) / 2 cos^2(2x) = (1 + cos(4x)) / 2

Now, substituting these identities into the original expression:

dx / [(1 - cos(4x)) / 2 * (1 + cos(4x)) / 2]

Simplifying further, we can multiply by the reciprocal of the denominator:

dx * (2 / [(1 - cos(4x)) * (1 + cos(4x))])

To continue simplifying, we can use the difference of squares identity:

a^2 - b^2 = (a + b)(a - b)

Here, let's consider a = 1 and b = cos(4x):

dx * (2 / [(1 - cos(4x)) * (1 + cos(4x))]) = dx * (2 / (1 - cos^2(4x)))

Now, using the identity cos^2(θ) + sin^2(θ) = 1, we can replace cos^2(4x) with sin^2(4x):

dx * (2 / (1 - sin^2(4x)))

Finally, using the identity 1 - sin^2(θ) = cos^2(θ), we can simplify further:

dx * (2 / cos^2(4x))

Therefore, the simplified expression is dx / cos^2(4x).