Mathematics

Evaluate : $$\displaystyle \int { \sqrt { 3{ x }^{ 2 }+x+1 }  } dx$$


SOLUTION
$$I=\int{\sqrt{3(x^2+\cfrac{x}{3}+\cfrac{1}{3})}}dx=\sqrt3\int{\sqrt{(x+\cfrac{1}{6})^2-\cfrac{1}{36}+\cfrac{1}{3}}}\\ \quad=\sqrt3\int{\sqrt{(x+\cfrac{1}{6})^2+(\cfrac{11}{6})^2}}dx\\ =\sqrt3[(\cfrac{x+1/6}{2})\sqrt{x^2+\cfrac{x}{3}+\cfrac{1}{3}}+\cfrac{11}{36}ln(x+\cfrac{1}{6}+\sqrt{x^2+\cfrac{x+1}{3}})]+C$$
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 Subjective Medium
Prove that $$\displaystyle\int^{\tan x}_{1/e}\dfrac{t}{1+t^2}dt+\displaystyle\int^{\cot x}_{1/e}\dfrac{1}{t(1+t^2)}dt=1$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
Evaluate: $$\displaystyle\int \dfrac {dx}{\sqrt {-x^2-x}} $$
  • A. $$\sin^{-1} (4x+1)+K$$
  • B. $$\sin^{-1} (4x-1)+K$$
  • C. None of these
  • D. $$\sin ^ {-1}(2x+1)+K$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
The value of $$\displaystyle \int_{ 1 }^{\frac{1+\sqrt{5}}{2}}\frac{x^2+1}{x^4-x^2+1}\log \left ( 1+x-\frac{1}{x} \right )\:dx$$
  • A. $$\displaystyle \frac{\pi}{2}\log_{e}2$$
  • B. $$\displaystyle -\frac{\pi}{2}\log_{e}2$$
  • C. noneof these
  • D. $$\displaystyle \frac{\pi}{8}\log_{e}2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
$$\displaystyle \int \dfrac{x \,\ell nx}{(x^2 - 1)^{3/2}}dx$$ equals
  • A. $$\sec^{-1} x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$
  • B. $$\cos^{-1} x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$
  • C. $$\sec x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$
  • D. $$arc \,\sec x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
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 $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer