Mathematics

# If $\int^{a}_{1} (3x^2+2x+1)dx=11$, find real values of a.

##### SOLUTION
We know,

$\displaystyle \int^{a}_{1}(3x^2+2x+1)dx$

we know that $\displaystyle\int{{x}^{n}dx}=\dfrac{{x}^{n+1}}{n+1}+c$

$\Rightarrow [x^3+x^2+x]^{a}_{1}=11$

$\Rightarrow (a^3+a^2+a)-(1+1+1)=11$

$\Rightarrow a^3+a^2+a-3=11$

$\Rightarrow a^3+a^2+a-14=0$

$\Rightarrow (a-2)(a^2+3a+7)=0$

since $a^2+3a+7$ does not have real values as its $D=3^2-4(2)(7)=-47<0$

$\therefore a=2$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 One Word Medium
$\displaystyle I= \int_{0}^{\pi /2}\frac{x\sin x\cos x}{\cos ^{4}x+\sin ^{4}x}dx$
$\displaystyle \therefore I= \pi ^{2}/k.$
what is k?

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $I= \displaystyle \int_{-2}^{2}\displaystyle \frac{e^{x}}{e^{x}+e^{-x}}\: dx$ then $I$ is equal to?
• A. $4$
• B. $3$
• C. $0$
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of $\int \dfrac {10^{x/2}}{\sqrt {10^{-x} - 10^{x}}}\, dx$ is
• A. $2\sqrt {10^{-x} + 10^{x}} + c$
• B. $\dfrac {1}{\log_{e}10}\sin h^{-1} (10^{x}) + c$
• C. $\dfrac {-1}{\log_{e}10}\sin h^{-1}(10^{x}) + c$
• D. $\dfrac {1}{\log_{2}10}\sin^{-1}(10^{x}) + c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int\frac{dx}{x\sqrt{1-x^{3}}}=$
• A. $\dfrac{1}{3}\log|\displaystyle \frac{\sqrt{1-x^{2}}+1}{\sqrt{1-x^{2}}-1}|+c$
• B. $\dfrac{1}{3}\log|\displaystyle \frac{1}{\sqrt{1-x^{3}}}|+c$
• C. $\displaystyle \frac{1}{3}\log|1-x^{3}|+c$
• D. $\dfrac{1}{3}\log|\displaystyle \frac{\sqrt{1-x^{3}}-1}{\sqrt{1-x^{3}}+1}|+c$

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 $\displaystyle f(x)>0 \forall x \in R$