Mathematics

# Evaluate:$\displaystyle\int{\dfrac{x\,dx}{a+bx}}$

##### SOLUTION
$\displaystyle\int{\dfrac{x\,dx}{a+bx}}$
$=\displaystyle\int{\dfrac{\left(a+bx-a\right)\,dx}{a+bx}}$
$=\displaystyle\int{\dfrac{\left(a+bx\right)\,dx}{a+bx}}-\displaystyle\int{\dfrac{a\,dx}{a+bx}}$
$=\displaystyle\int{dx}-\dfrac{a}{b}\displaystyle\int{\dfrac{b\,dx}{a+bx}}$
Let $t=a+bx\Rightarrow\,dt=b\,dx$
$=\displaystyle\int{dx}-\dfrac{a}{b}\displaystyle\int{\dfrac{dt}{t}}$
$=\displaystyle\int{dx}-\dfrac{a}{b}\ln{\left|t\right|}+c$
$=x-\dfrac{a}{b}\ln{\left|\left(a+bx\right)\right|}+c$

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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 Hard
If $\displaystyle f\left ( x \right )$ and $\displaystyle g\left ( x \right )$ be continuous functions over the closed interval $\displaystyle \left [ 0, a \right ]$ such that $\displaystyle f\left ( x \right )= f\left ( a-x \right )$ and $\displaystyle g\left ( x \right )+g\left ( a-x \right )= 2.$ Then $\displaystyle \int_{0}^{a}f\left (x \right )\dot g\left (x \right )dx$ is equal to
• A. $\displaystyle \int_{0}^{a}g\left ( x \right )dx$
• B. $\displaystyle 2a$
• C. none of these
• D. $\displaystyle \int_{0}^{a}f\left ( x \right )dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $I=\int \dfrac{\sin2x}{(3+4\cos x)^3}dx$ then $I$ is equal to
• A. $\dfrac{3 \cos x+8}{(3+4 \cos x)^2}+C$
• B. $\dfrac{3 \cos x+8}{16(3+4 \cos x)^2}+C$
• C. $\dfrac{3+ \cos x}{(3+4 \cos x)^2}+C$
• D. $\dfrac{3+8 \cos x}{16(3+4 \cos x)^2}+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \lim_{n \rightarrow \infty} \left[\displaystyle \frac{\sqrt{n^{2}-1^{2}}}{n^{2}}+\frac{\sqrt{n^{2}-2^{2}}}{n^{2}}+\frac{\sqrt{n^{2}-3^{2}}}{n^{2}}+\ldots.n terms\right]=$
• A. $\displaystyle \frac{\pi}{2}$
• B. $\displaystyle \frac{\pi}{3}$
• C. $\displaystyle \frac{2\pi}{4}$
• D. $\displaystyle \frac{\pi}{4}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium

$\displaystyle \overset{\infty}{\underset{0}{\int}} x^6 e^{\tfrac{-x}{2}} dx =$

• A. $2^6 \left \lfloor 6 \right.$
• B. $\dfrac{ \left \lfloor 6 \right.}{2^7}$
• C. $\dfrac{ \left \lfloor 6 \right.}{2^6}$
• D. $2^7 \left \lfloor 6 \right.$

Given that for each $\displaystyle a \in (0, 1), \lim_{h \rightarrow 0^+} \int_h^{1-h} t^{-a} (1 -t)^{a-1}dt$ exists. Let this limit be $g(a)$
In addition, it is given that the function $g(a)$ is differentiable on $(0, 1)$