Mathematics

# Evaluate using limit of sum:$\displaystyle \int_{1}^{3} {(x+1)^2}dx$

$28$

##### SOLUTION

$\displaystyle \int_{1}^{3} {(x+1)^2}dx$

Let $t=(x+1)\implies dt=dx$

$x \to 1\to 3$

$t\to 2 \to 4$

$\displaystyle \int _2^4 t^2 dt$

$\left.\dfrac {t^3}3\right]_2^4$

$\dfrac{4^3}{2}-\dfrac {2^3}{2}$

$32-4=28$

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Single Correct Hard
The integral $\displaystyle \int x { \cos^{ -1 }\left(\displaystyle \frac { 1-x^{ 2 } }{ 1+x^{ 2 } } \right) dx }$ is equal to :
(Note : $(x>0)$)
• A. $-x + (1+x^2) \cot^{-1} x+c$
• B. $-x-(1+x^{2})\tan^{-1}xc$
• C. $x-(1+x^2)\cot^{-1}x+c$
• D. $-x+(1+x^{2})\tan^{-1}x+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Matrix Hard
The antiderivative of

 $\displaystyle \frac{\sec x}{\left ( \sec x+\tan x \right )^{2}}$ $\displaystyle \log\left | \frac{\sin x-2}{\sin x-1} \right |+C$ $\displaystyle \frac{\cos x}{\left ( \sin x-1 \right )\left ( \sin x-2 \right )}$ $\displaystyle -\frac{1}{2}\left ( \sec x+\tan x \right )^{-2}+C$ $\displaystyle \sin ^{-1}\frac{2x}{1+x^{2}}$ $\displaystyle 2x\tan^{-1}x-log\left ( 1+x^{2} \right )+C$ $\displaystyle \sqrt{\tan x}+\sqrt{\cot x}$ $\displaystyle \sqrt{2}\tan ^{-1}\left ( \frac{\tan x-1}{\sqrt{2\tan x}} \right )$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\displaystyle \int_0^1e^{2-3x}dx$ as a limit of a sum.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\int { \cfrac { dx }{ \sqrt { { x }^{ 2 }+6x+5 } } } =$?
• A. $\log { \left| x+\sqrt { { x }^{ 2 }+6x+5 } \right| } +C\quad$
• B. $\log { \left| x-\sqrt { { x }^{ 2 }+6x+5 } \right| } +C$
• C. none of these
• D. $\log { \left| \left( x+3 \right) +\sqrt { { x }^{ 2 }+6x+5 } \right| } +C$

Evaluate $\int { \dfrac { dx }{ \sqrt { 5{ x }^{ 2 }-2x } } }$