Mathematics

# $I = \int \frac { 1 } { \sqrt { 2 x ^ { 2 } + 3 x + 8 } } d x$

##### SOLUTION
$I=\int { \frac { 1 }{ \sqrt { 2{ x }^{ 2 }+3x+8 } } dx } \\ =\frac { 1 }{ \sqrt { 2 } } \int { \frac { 1 }{ \sqrt { { x }^{ 2 }+\frac { 3 }{ 2 } x+4 } } dx } \\ =\frac { 1 }{ \sqrt { 2 } } \int { \frac { 1 }{ \sqrt { { x }^{ 2 }+2\times \frac { 3 }{ 4 } x+\frac { 9 }{ 16 } +4-\frac { 9 }{ 16 } } } dx } \\ =\frac { 1 }{ \sqrt { 2 } } \int { \frac { 1 }{ \sqrt { { \left( x+\frac { 3 }{ 4 } \right) }^{ 2 }+\frac { 55 }{ 16 } } } dx } \\ =\frac { 1 }{ \sqrt { 2 } } \int { \frac { 1 }{ \sqrt { { \left( x+\frac { 3 }{ 4 } \right) }^{ 2 }+{ \left( \frac { \sqrt { 55 } }{ 4 } \right) }^{ 2 } } } dx }$
let $x+\frac { 3 }{ 4 } =u$
$dx=du$
so $I=\frac { 1 }{ \sqrt { 2 } } \int { \frac { 1 }{ \sqrt { { u }^{ 2 }+{ \left( \frac { \sqrt { 55 } }{ 4 } \right) }^{ 2 } } } du } \\ =\frac { 1 }{ \sqrt { 2 } } \ln { \left[ u+\sqrt { { u }^{ 2 }+{ \left( \frac { \sqrt { 55 } }{ 4 } \right) }^{ 2 } } \right] } +c$
putting $u=x+\frac { 3 }{ 4 }$
$I=\frac { 1 }{ \sqrt { 2 } } \ln { \left[ x+\frac { 3 }{ 4 } +\sqrt { { \left( x+\frac { 3 }{ 4 } \right) }^{ 2 }+{ \left( \frac { \sqrt { 55 } }{ 4 } \right) }^{ 2 } } \right] } +c\\ =\frac { 1 }{ \sqrt { 2 } } \ln { \left[ x+\frac { 3 }{ 4 } +\sqrt { { x }^{ 2 }+\frac { 3 }{ 2 } x+\frac { 9 }{ 16 } +\frac { 55 }{ 16 } } \right] } +c\\ =\frac { 1 }{ \sqrt { 2 } } \ln { \left[ x+\frac { 3 }{ 4 } +\sqrt { { x }^{ 2 }+\frac { 3 }{ 2 } x+4 } \right] } +c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
Evaluate $\displaystyle \int \dfrac{1}{1+3\sin^{2}{x}+8\cos^{2}{x}}dx$
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• C. $None\ of\ these$
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solve the following.
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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.