Integrals Test

Result

Integrals Test - 65

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Weekly Quiz Competition

Question 1
1 / -0

If $$I_n=\displaystyle\int^1_0\dfrac{dx}{(1+x^2)^n}; n\in N$$, then which of the following statements hold good?

A
$$2n I_{n+1}=2^{-n}+(2n-1)I_n$$

B
$$I_2=\dfrac{\pi}{8}+\dfrac{1}{4}$$

C
$$I_2=\dfrac{\pi}{8}-\dfrac{1}{4}$$

D
$$I_3=\dfrac{\pi}{16}-\dfrac{5}{48}$$

Solution
Question 2
1 / -0

$$\int _ { 0 } ^ { 2 } \cfrac { \sqrt { x } } { \sqrt { x } + \sqrt { 2 - x } }$$ is equal to

A
$$-1$$

B
$$0$$

C
$$2$$

D
$$1$$

Solution
Question 3
1 / -0

$$\int _{ 0 }^{ \infty }{ \frac { { x }^{ 2 }+1 }{ { x }^{ 4 }+{ 7x }^{ 2 }+1 } } $$ dx=

A
$$\pi $$

B
$$\frac { \pi }{ 2 } $$

C
$$\frac { \pi }{ 3 } $$

D
$$\frac { \pi }{ 6 } 4$$

Solution
Question 4
1 / -0

The solution of $$x$$ of the equation $$\displaystyle \int_{\sqrt{2}}^{x}{\dfrac{dt}{t\sqrt{t^{2}-1}}}=\dfrac{\pi}{2}$$ is

A
$$2\sqrt{2}$$

B
$$2$$

C
$$\pi$$

D
$$-\sqrt{2}$$

Solution
Question 5
1 / -0

Find the value of the equation : $$\int _ { \ln \lambda } ^ { \ln \left( \frac { 1 } { \lambda } \right) } \dfrac { f \left( \dfrac { x ^ { 2 } } { 3 } \right) ( f ( x ) + f ( - x ) ) } { g \left( 3 x ^ { 2 } \right) ( g ( x ) - g ( - x ) ) } d x =?$$ where $$\lambda > 1$$

A
$$0$$

B
$$1$$

C
$$\lambda$$

D
$$1 / \lambda$$

Solution
Question 6
1 / -0

Find the value of the equation $$\int _ { 0 } ^ { \infty } \dfrac { d x } { \left( x + \sqrt { x ^ { 2 } + 1 } \right) ^ { 3 } } =?$$

A
$$3 / 8$$

B
$$1/8$$

C
$$-3/8$$

D
$$-1/8$$

Solution
Question 7
1 / -0

The value of $$\displaystyle \int _{0}^{1}\dfrac{x^{4}(1-x)^{4}}{1+x^{2}}\ dx$$ is

A
$$0$$

B
$$\dfrac{2}{105}$$

C
$$\dfrac{22}{7}-\pi$$

D
$$\dfrac{71}{15}-\dfrac{3\pi}{2}$$

Solution
Question 8
1 / -0

If $$u _ { n } = \int _ { 0 } ^ { \pi / 2 } x ^ { n } \sin x \quad d x$$ then the value of $$u _ { 10 } + 90 \mathrm { u } _ { 8 }$$ is:

A
$$9$$$$\left( \frac { \pi } { 2 } \right) ^ { 8 }$$

B
$$\left( \frac { \pi } { 2 } \right) ^ { 9 }$$

C
$$10$$$$\left( \frac { \pi } { 2 } \right) ^ { 9 }$$

D
none of these

Solution
Question 9
1 / -0

$$\int^{2/3}_0 \dfrac{dx}{4+9x^2}$$ equals

A
$$\dfrac{\pi}{6}$$

B
$$\dfrac{\pi}{12}$$

C
$$\dfrac{\pi}{24}$$

D
$$\dfrac{\pi}{4}$$
Question 10
1 / -0

$$\int _{ -1 }^{ 1/2 }{ \dfrac { { e }^{ x }\left( 2-{ x }^{ 2 } \right) dx }{ \left( 1-x \right) \sqrt { 1-{ x }^{ 2 } } } }$$ is equal to

A
$$\dfrac { \sqrt { e } }{ 2 } \left( \sqrt { 3 } +1 \right)$$

B
$$\dfrac{ \sqrt { 3e } }{ 2 }$$

C
$$\sqrt{ 3e }$$

D
$$\sqrt{ \dfrac { e }{ 3 } }$$

Solution