Mathematics

Evaluate $$\displaystyle\int_{2}^{4}\dfrac{x}{x^{2}+1}dx$$


SOLUTION
$$\displaystyle\int_{4}^{2}{\dfrac{x}{{x}^{2}+1}dx}$$

$$=\dfrac{1}{2} \displaystyle\int_{4}^{2}{\dfrac{2x}{{x}^{2}+1}dx}$$   

Let $$t={x}^{2}+1\Rightarrow\,dt=2x\,dx$$

$$=\dfrac{1}{2} \displaystyle\int{\dfrac{dt}{t}}$$ 

$$=\dfrac{1}{2}\left[\log{\left|t\right|}\right]$$

$$=\dfrac{1}{2}\left[\log{\left|{x}^{2}+1\right|}\right]_{4}^{2}$$ where $$t={x}^{2}+1$$

$$=\dfrac{1}{2}\left[\log{\left|{2}^{2}+1\right|}-\log{\left|{4}^{2}+1\right|}\right]$$

$$=\dfrac{1}{2}\left[\log{\left|4+1\right|}-\log{\left|16+1\right|}\right]$$

$$=\dfrac{1}{2}\left[\log{\left|5\right|}-\log{\left|17\right|}\right]$$

$$=\dfrac{1}{2}\log{\left|\dfrac{5}{17}\right|}$$

View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Single Correct Hard
Resolve $$\displaystyle \frac{(2x^2+3)(x^2+4)}{(3x^2+1)(5x^2+3)}$$ into partial fractions.
  • A. $$\displaystyle \frac{2}{15}+\frac{77}{12(3x^2+1)}-\frac{153}{10(5x^2+3)}$$
  • B. $$\displaystyle \frac{2}{15}-\frac{77}{12(3x^2+1)}+\frac{153}{20(5x^2+3)}$$
  • C. $$\displaystyle \frac{2}{15}-\frac{77}{12(3x^2+1)}+\frac{153}{10(5x^2+3)}$$
  • D. $$\displaystyle \frac{2}{15}+\frac{77}{12(3x^2+1)}-\frac{153}{20(5x^2+3)}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate $$\int \int_{R} e^{-(x^{2} + y^{2})}dx dy$$, where $$R$$ is the region bounded by the circle $$x^{2} + y^{2} = a^{2}$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
$$\displaystyle \int { { e }^{ x }\cos ^{ 2 }{ x } dx } $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
Let $$\displaystyle I = \int_{1}^{3}\sqrt{x^4 + x^2}$$ dx , then
  • A. $$\displaystyle I > 6\sqrt{10}$$
  • B. $$\displaystyle 2\sqrt{2} < I < 6\sqrt{10}$$
  • C. $$\displaystyle I < 1$$
  • D. $$\displaystyle I < 2\sqrt{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
The value of $$\displaystyle \int_0^{\dfrac{\pi}{2}}\dfrac{\sqrt{\cot t}}{\sqrt{\cot t}+\sqrt{\tan t}}dt$$
  • A. $$\dfrac{\pi}{2}$$
  • B. $$\dfrac{\pi}{6}$$
  • C. $$\dfrac{\pi}{8}$$
  • D. $$\dfrac{\pi}{4}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer