Mathematics

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

##### SOLUTION
$I=\int{\sqrt{3(x^2+\cfrac{x}{3}+\cfrac{1}{3})}}dx=\sqrt3\int{\sqrt{(x+\cfrac{1}{6})^2-\cfrac{1}{36}+\cfrac{1}{3}}}\\ \quad=\sqrt3\int{\sqrt{(x+\cfrac{1}{6})^2+(\cfrac{11}{6})^2}}dx\\ =\sqrt3[(\cfrac{x+1/6}{2})\sqrt{x^2+\cfrac{x}{3}+\cfrac{1}{3}}+\cfrac{11}{36}ln(x+\cfrac{1}{6}+\sqrt{x^2+\cfrac{x+1}{3}})]+C$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### 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

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

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

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

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