Integrals Test

Result

Integrals Test - 62

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Weekly Quiz Competition

Question 1
1 / -0

The value of the definite integral $$\int _{ 2 }^{ 3 }{ \left[ \sqrt { 2x-\sqrt { 5\left( 4x-5 \right) } } +\sqrt { 2x+\sqrt { 5\left( 4x-5 \right) } } \right] } dx=$$

A
$$\dfrac {7\sqrt {3}+3\sqrt {5}}{3\sqrt {2}}$$

B
$$4\sqrt {2}$$

C
$$4\sqrt {3}+\dfrac {4}{3}$$

D
$$\dfrac {7\sqrt {7}-2\sqrt {5}}{3\sqrt {2}}$$

Solution
Question 2
1 / -0

The integral $$\displaystyle \int_{\dfrac{\pi}{12}}^{\dfrac{\pi}{4}}{\dfrac{8\cos 2x}{(\tan x+\cot x)^{3}}dx}$$ equals :

A
$$\dfrac{15}{128}$$

B
$$\dfrac{15}{64}$$

C
$$\dfrac{13}{32}$$

D
$$\dfrac{13}{256}$$

Solution
Question 3
1 / -0

Let $$f\left(x\right)+f\left(\dfrac{1}{x}\right)=F\left(x\right)$$ where $$f\left(x\right)=\displaystyle\int_{1}^{x}{\dfrac{\ln{t}}{1+t}dx}$$.Then $$F\left(e\right)=$$

A
$$\dfrac{-1}{2}$$

B
$$\dfrac{1}{2}$$

C
$$1$$

D
$$-1$$

Solution
Question 4
1 / -0

The value of $$\displaystyle \int _{0}^{\infty}\dfrac{\sqrt{x^2+1}}{(x+\sqrt{x^{2}+1})^{n+1}} .dx \ \forall \ n\ \in \ N- \{ \pm 1 \}$$ is

A
$$0$$

B
$$n(n^{2}-1)$$

C
$$\dfrac{n}{(n^{2}-1)}$$

D
$$n^{2}$$

Solution
Question 5
1 / -0

$$\displaystyle \int_{-3\pi}^{3\pi}{\sin^{2}\theta\sin^{2} 2\theta d \theta}$$ is equal to-

A
$$\pi$$

B
$$\dfrac{3\pi}{2}$$

C
$$\dfrac{5\pi}{2}$$

D
$$6\pi$$

Solution
Question 6
1 / -0

If $$P=\displaystyle \lim _{ n\rightarrow \infty }{ \frac { { \left( \prod _{ r=1 }^{ n }{ \left( { n }^{ 3 }+{ r }^{ 3 } \right) } \right) }^{ 1/n } }{ { n }^{ 3 } } }$$ and $$\lambda =\displaystyle \int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 3 } } } $$ then $$In P$$ is equal to

A
$$In 2-1+\lambda$$

B
$$In 2-3+3\lambda$$

C
$$2In 2-\lambda$$

D
$$In 4-3+3\lambda$$

Solution
Question 7
1 / -0

Let $$f:R\longrightarrow R,g : R\longrightarrow R$$ be continuous functions. then the value of integeral.
$$\int _{ \ell n\lambda }^{ \ell n/\lambda }{ \frac { f\left( \dfrac { { x }^{ 2 } }{ 4 } \right) \left[ f\left( x \right) -f\left( -x \right) \right] }{ g\left( \dfrac { { x }^{ 2 } }{ 4 } \right) \left[ g\left( x \right) +g\left( -x \right) \right] } } dx$$ is:

A
depend on $$\lambda$$

B
a non-zero constant

C
zero

D
1
Question 8
1 / -0

$$\displaystyle \int_{0}^{\pi/2}{(\sin x-\cos x).\log(\sin x+\cos x)dx}$$=

A
$$\dfrac{\pi}{4}$$

B
$$\dfrac{\pi}{2}$$

C
$$0$$

D
$$2$$

Solution
Question 9
1 / -0

$$\int^2_0 (\sqrt{\dfrac{4-x}{x}} - \sqrt{\dfrac{x}{4-x}})dx$$ is equal to

A
$$0$$

B
$$8$$

C
$$4$$

D
$$16$$

Solution
Question 10
1 / -0

Evaluate:
$$\displaystyle\int_{-2}^{3}{\left|1-{x}^{2}\right|dx}$$

A
$$\dfrac{28}{3}$$

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

C
$$\dfrac{8}{3}$$

D
$$\dfrac{5}{3}$$

Solution