Mathematics

# Evaluate the following:$\displaystyle \int (3x + 1)\sqrt{2x - 1}dx$

##### SOLUTION

$I=\int { \left( 3x+1 \right) \sqrt { 2x-1 } } dx\\ Put\quad 2x-1=t\\ 2dx=dt\\ dx=\cfrac { 1 }{ 2 } dt\\ \therefore 3x+1=\left( t+1 \right) \cfrac { 3 }{ 2 } +1\\ \therefore I=\cfrac { 1 }{ 2 } \int { \left( \cfrac { 3 }{ 2 } t+\cfrac { 3 }{ 2 } +1 \right) \sqrt { t } dt } \\ I=\cfrac { 1 }{ 2 } \int { \left( \cfrac { 3 }{ 2 } { t }^{ \cfrac { 3 }{ 2 } }+\cfrac { 5 }{ 2 } { t }^{ \cfrac { 1 }{ 2 } } \right) dt } \\ I=\cfrac { 1 }{ 2 } \left( \cfrac { 3 }{ 2 } \times \cfrac { 2 }{ 5 } { t }^{ \cfrac { 5 }{ 2 } }+\cfrac { 5 }{ 2 } \times \cfrac { 2 }{ 3 } { t }^{ \cfrac { 3 }{ 2 } } \right) +C\quad \left( \because \int { { x }^{ n }dx=\cfrac { 1.{ x }^{ n+1 } }{ n+1 } } \right) \\ I=\cfrac { 3 }{ 10 } { t }^{ \cfrac { 5 }{ 2 } }+\cfrac { 5 }{ 6 } { t }^{ \cfrac { 3 }{ 2 } }+C\\ I=\cfrac { 3 }{ 10 } { \left( 2x-1 \right) }^{ ^{ \cfrac { 5 }{ 2 } } }+\cfrac { 5 }{ 6 } { \left( 2x-1 \right) }^{ \cfrac { 3 }{ 2 } }+C$

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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 $\displaystyle \int _{ \pi /4 }^{ 3\pi /4 }{ \frac { x\sin { x } }{ 1+\sin { x } } } dx$
• A. $\displaystyle \frac { { \pi }^{ 2 } }{ 4 } -\left( \sqrt { 2 } +1 \right) \pi$
• B. $\displaystyle \frac { { \pi }^{ 2 } }{ 2 } -\left( \sqrt { 2 } -1 \right) \pi$
• C. $\displaystyle \frac { { \pi }^{ 2 } }{ 2 } -\left( \sqrt { 2 } +1 \right) \pi$
• D. $\displaystyle \frac { { \pi }^{ 2 } }{ 4 } -\left( \sqrt { 2 } -1 \right) \pi$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the given integral.
$\displaystyle\int{{e}^{\log{\sqrt{x}}}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of $\displaystyle \int_{ {\pi^{3}}/{27}}^{ {\pi^{3}}/{8}}sinx.dt$ , where $t= x^3$, is?
• A. $cos\displaystyle \frac{\pi^{3}}{27}-cos\displaystyle \frac{\pi^{3}}{8}$
• B. $\displaystyle \frac{\pi^2}{6}$
• C. None of these
• D. $\displaystyle \frac{\pi^{2}}{6}+(3-\sqrt3)\pi-3$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \frac{3x-1}{(1-x+x^{2})(2+x)}=$
• A. $\displaystyle \frac{x}{x^{2}-x+1}+\frac{1}{x+2}$
• B. $\displaystyle \frac{x}{x^{2}+x+1}+\frac{2}{x+2}$
• C. $\displaystyle \frac{x}{ -x+1}-\frac{2}{x+2}$
• D. $\displaystyle \frac{x}{x^{2}-x+1}-\frac{1}{x+2}$

$(x^2+1) \log x$