Mathematics

Evaluate the following integral:
$$\int { \cfrac { 4x+3 }{ \sqrt { 2{ x }^{ 2 }+3x+1 }  }  } dx$$


SOLUTION
Let 
$$t=2{x}^{2}+3x+1\Rightarrow\,dt=\left(4x+3\right)dx$$

$$\displaystyle\int{\dfrac{\left(4x+3\right)dx}{\sqrt{2{x}^{2}+3x+1}}}$$

$$=\displaystyle\int{\dfrac{dt}{\sqrt{t}}}$$

$$=\displaystyle\int{{t}^{\frac{-1}{2}}dt}$$

$$=\dfrac{{t}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+c$$

$$=\dfrac{{t}^{\frac{1}{2}}}{\dfrac{1}{2}}+c$$

$$=2\sqrt{t}+c$$

$$=2\sqrt{2{x}^{2}+3x+1}+c$$ where $$t=2{x}^{2}+3x+1$$

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 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
Evaluate:$$\displaystyle\int \frac{x}{\sqrt{9-x^{4}}}dx$$
  • A. $$ \sin^{-1}\left ( \dfrac{x^{2}}{3} \right )+C$$
  • B. $$\dfrac{1}{2} \sin^{-1}\left ( \dfrac{x^{2}}{9} \right )+C$$
  • C. none of these
  • D. $$\dfrac{1}{2} \sin^{-1}\left ( \dfrac{x^{2}}{3} \right )+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Hard
$$\int { \dfrac { \tan { 2\theta  }  }{ \sqrt { \cos ^{ 6 }{ \theta  } +\sin ^{ 6 }{ \theta  }  }  } d\theta  }$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
$$\dfrac{2x^3+x^2-5}{x^4-25}=\dfrac{Ax+B}{x^2-5}+\dfrac{Cx+1}{x^2+5}\Rightarrow (A, B, C)=$$
  • A. (1, 1, 1)
  • B. (1, 1, 0)
  • C. (1, 2, 1)
  • D. (1, 0, 1)

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
If $$\int \dfrac {1}{x \sqrt {1 - x^{3}}}dx = a\log \left |\dfrac {\sqrt {1 - x^{3}} - 1}{\sqrt {1 - x^{3}} + 1}\right | + b$$ then $$a =$$
  • A. $$\dfrac {2}{3}$$
  • B. $$-\dfrac {2}{3}$$
  • C. None of these
  • D. $$\dfrac {1}{3}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
$$\displaystyle \int_2^3\dfrac{dx}{x^2-1}$$
  • A. $$log\dfrac{3}{2}$$
  • B. $$2log\dfrac{3}{2}$$
  • C. $$\log\dfrac{3}{2}$$
  • D. $$\dfrac{1}{2}log\dfrac{3}{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer