Mathematics

# Evaluate the following integrals:$\int { \cfrac { \sin { 8x } }{ \sqrt { 9+\sin ^{ 4 }{ 4x } } } } dx$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
Evaluate the following integral
$\int { \cfrac {2\cos { 4x } -\cos { 2x } }{ \sin { 4x } -\sin { 2x } } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Evaluate: $\int _{ a }^{ b }{ x } dx$ using limit of sum.
• A. $\dfrac{b^2-a^2}{3}$
• B. $\dfrac{b^2+a^2}{2}$
• C. None of these
• D. $\dfrac{b^2-a^2}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int _{ 0 }^{ \pi /4 }{ \tan ^{ 2 }{ x } dx= }$
• A. $1+\dfrac {\pi}{4}$
• B. $\dfrac {-\pi}{4}-1$
• C. $\dfrac {\pi}{4}-1$
• D. $1-\dfrac {\pi}{4}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If  $\displaystyle \int_{0}^{k}\frac{\cos x}{1+\sin^{2}x}dx=\frac{\pi}{4}$ then ${k}=?$
• A. $1$
• B. $\pi/4$
• C. $\pi/6$
• D. $\pi/2$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard

In 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.