Mathematics

# Let g be continuous function on R such that $\int g(x)\mathrm{d} x=f(x)+C$, where C is constant of integration. If f(x) is an odd function, $f(1)=3$ and $\int_{-1}^{1}f^{2}(x)g(x)\mathrm{d} x=\lambda$, then $\frac{\lambda }{2}$ is equal to

9

##### SOLUTION
$\int g(x){ d }x=f(x)+C$
$\Rightarrow g(x)=f'(x)$

Now,$\int _{ -1 }^{ 1 } f^{ 2 }(x)g(x){ d }x=\lambda$
Applying integration by parts,

$\Rightarrow { [f^{ 2 }(x)f(x)] }_{ -1 }^{ 1 }-\int _{ -1 }^{ 1 }{ 2f(x)f'(x)f(x)dx } =\lambda$

$\Rightarrow { [f^{ 3 }(x)] }_{ -1 }^{ 1 }-\int _{ -1 }^{ 1 }{ 2{ f }^{ 2 }(x)g(x)dx } =\lambda$

$\Rightarrow { [f(1)] }^{ 3 }-{ [f(-1)] }^{ 3 }=3\lambda$

$\Rightarrow 27+27=3\lambda$ ( $\because$ f(x) is an odd function)

$\Rightarrow \displaystyle \frac{\lambda}{2}=9$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Medium
Let $f(x)$ and $g(x)$ be two function satisfying $f(x^2)+g (4-x)=4x^3, g(4-x)+g(x)=0$, then the value of $\int_{-4}^{4} f(x^2)dx$ is:
• A. 64
• B. 256
• C.
• D. 512

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Find : $\displaystyle \int_0^1 \dfrac{ x}{1+x^{2}} d x$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The value of $\int \dfrac {dx}{x(\log x)^{m}}$ is
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• B. $\dfrac {(\log x)^{m - 1}}{m - 1} + C$
• C. $\dfrac {(\log x)^{1 - m}}{m} + C$
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1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
The value of $\int \sqrt { \dfrac { e^ x\quad -\quad 1 }{ e^ x\quad +\quad 1 } } dx$ is equal to
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• D. $\ell n\left( e^ x+\sqrt { e^ 2x\quad -1 } \right) -sin^ -1(e^ {-x})+c$

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