**Mathematics**

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Evaluate the following integrals:

$$\int { \cfrac { \sin { 8x } }{ \sqrt { 9+\sin ^{ 4 }{ 4x } } } } dx$$

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$$\int { \cfrac {2\cos { 4x } -\cos { 2x } }{ \sin { 4x } -\sin { 2x } } } dx\quad $$

**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

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**A.**$$\dfrac{b^2-a^2}{3}$$ -
**B.**$$\dfrac{b^2+a^2}{2}$$ -
**C.**None of these -
**D.**$$\dfrac{b^2-a^2}{2}$$

**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

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**A.**$$1+\dfrac {\pi}{4}$$ -
**B.**$$\dfrac {-\pi}{4}-1$$ -
**C.**$$\dfrac {\pi}{4}-1$$ -
**D.**$$1-\dfrac {\pi}{4}$$

**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

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**A.**$$1$$ -
**B.**$$\pi/4$$ -
**C.**$$\pi/6$$ -
**D.**$$\pi/2$$

**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

View AnswerIn calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

$$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$$ $$\int\, u^{n}(x)v_{n}(x)\, dx$$ where $$v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$$

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when calculating $$\int P_{n}(x)\, Q(x)\, dx$$, where $$P_{n}(x)$$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.

**Asked in: **Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

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