Mathematics

# Evaluate the following integrals:$\int { \sqrt { 2ax-{ x }^{ 2 } } } dx\quad$

##### SOLUTION
Let $I=\displaystyle\int \sqrt{2 a x-x^2}\ d x$

$\implies I=\displaystyle\int \sqrt{-(-2(a)(x)+x^2)}\ d x$

$\implies I=\displaystyle\int \sqrt{a^2-(a^2-2(a)(x)+x^2}\ d x$

$\implies I=\displaystyle\int \sqrt{a^2-(a-x)^2}\ d x$

Put $t=a-x\implies d t=-d x$

$\implies I=-\displaystyle\int \sqrt{a^2-t^2}\ d t$

As we know that

$\displaystyle\int \sqrt{a^2-x^2}\ d x=\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{x}{a}\right)+C$

$I=-\dfrac{t}{2}\sqrt{a^2-t^2}-\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{t}{a}\right)+C$

$I=-\dfrac{a-x}{2}\sqrt{a^2-(a-x)^2}-\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{a-x}{a}\right)+C$

$I=\dfrac{1}{2}(x-a)\sqrt{2 a x-x^2}-\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{a-x}{a}\right)+C$

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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
Solve : $\displaystyle \int \, \dfrac{dx}{\sqrt{2x - x^2}}$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Medium
$\displaystyle \int \sec ^{2}x\log \left ( 1+\sin^{2}x \right )dx=\tan x\log \left ( 1+\sin ^{2}x \right )-2x+\sqrt{k}\tan^{-1}\sqrt{k}\tan x$. Find the value of $k$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
If n is any positive integer, show that the integral part of $( 3 + \sqrt7)^n$ is an odd number.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Let $\displaystyle F\left ( x \right )=f\left ( x \right )+f\left ( \frac{1}{x} \right )$ where $\displaystyle f\left ( x \right )=\int_{1}^{x}\frac{\log t}{1+t}dt$
Then $F(e)$ is equal to?
• A. $1$
• B. $2$
• C. $0$
• D. $1/2$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$