Mathematics

# The integral $\int _{ 0 }^{ \pi }{ \sqrt { 1+4\sin { ^{ 2 }\frac { x }{ 2 } -4 } \sin { \frac { x }{ 2 } } } dx }$ is equal to

$\pi-4$

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

#### Realted Questions

Q1 Subjective Medium
$\int {\sqrt {\dfrac{{1 - x}}{{1 + x}}} } \,dx =$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\displaystyle\int^1_0\dfrac{dx}{(1+x^2)}$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Value of $I= \displaystyle \int_{0}^{a}\displaystyle \frac{dx}{x+\sqrt{a^{2}-x^{2}}}$ is
• A. $\pi /2$
• B. $\pi$
• C. $2\pi$
• D. $\pi /4$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Evaluate: $\displaystyle \int \left [ \frac{1-\sqrt{x}}{1+\sqrt{x}} \right ]^{1/2}\frac{dx}{x}$
• A. $\displaystyle 2\sin ^{-1}\sqrt{x}+2\log \left [ \frac{1+\sqrt{1-x}}{\sqrt{x}} \right ]$
• B. $\displaystyle 2\cos ^{-1}\sqrt{x}+2\log \left [ \frac{1-\sqrt{1+x}}{\sqrt{x}} \right ]$
• C. $\displaystyle 2\sin ^{-1}\sqrt{x}-2\log \left [ \frac{1-\sqrt{1+x}}{\sqrt{x}} \right ]$
• D. $\displaystyle 2\cos ^{-1}\sqrt{x}-2\log \left [ \frac{1+\sqrt{1-x}}{\sqrt{x}} \right ]$

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.