Mathematics

Evaluate the definite integral :
$$\displaystyle \int_{0}^{1} \dfrac {2x+3}{5x^2 +1}dx$$


SOLUTION
$$I=\displaystyle \int_0^1 \dfrac {2x+3}{5x^2 +1}dx =\displaystyle \int_0^1 \dfrac {2x}{5x^2 +1}dx +\displaystyle \int_0^1 \dfrac {3}{5x^2 +1}dx =\dfrac {1}{5} \displaystyle \int_0^1 \dfrac {10x}{5x^2 +1}dx +3\displaystyle \int_0^1 \dfrac {1}{(\sqrt {5} x)^2 +1} dx $$
$$\Rightarrow \ I=\dfrac {1}{5}[\log (5x^2 +1)]_0^1 +\dfrac {3}{\sqrt 5}\left [\tan^{-1} \dfrac {\sqrt 5 x}{1}\right]_0^1$$
$$\Rightarrow \ I=\dfrac {1}{5}(\log 6-\log 1)+\dfrac {3}{\sqrt 5}\tan^{-1}\sqrt 5 =\dfrac {1}{5}\log 6+\dfrac {3}{\sqrt 5}\tan^{-1}\sqrt 5$$

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 Medium
Value of $$I=\int _{ 0 }^{ \frac { \pi  }{ 2 }  }{ \frac { { (\sin { x } +\cos { x } ) }^{ 2 } }{ \sqrt { 1+\sin { 2x }  }  } dx } $$ is 
  • A. $$1$$
  • B. $$2$$
  • C. $$4$$
  • D. $$0$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
$$\int \sin \frac { x } { 2 } d x$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
$$\displaystyle \int \sqrt {1+x \sqrt {1+(x+1) \sqrt {1+(x+2) (x+4)}}}$$ $$dx$$ is equal to
  • A. $$\displaystyle \frac{x^2}{2} - x+ c$$
  • B. $$\displaystyle \frac{x^2}{2} + c$$
  • C. $$x+c$$
  • D. $$\displaystyle \frac{x^2}{2} + x+ c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Write a value of 
$$\int { { e }^{ 2{ x }^{ 2 }+\ln { x }  } } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let $$n \space\epsilon \space N$$ & the A.M., G.M., H.M. & the root mean square of $$n$$ numbers $$2n+1, 2n+2, ...,$$ up to $$n^{th}$$ number are $$A_{n}$$, $$G_{n}$$, $$H_{n}$$ and $$R_{n}$$ respectively. 
On the basis of above information answer the following questions

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer