Integrals Test

Result

Integrals Test - 63

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Weekly Quiz Competition

Question 1
1 / -0

If $$2\displaystyle \int_{0}^{1}\tan^{-1}xdx=\displaystyle \int_{0}^{1}\cot^{-1}(1-x+x^{2})dx$$, then $$\displaystyle \int_{0}^{1}\tan^{-1}(1-x+x^{2})dx$$ is equal to:

A
$$\log 4$$

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

C
$$\log 2$$

D
$$\dfrac {\pi}{2}-\log 4$$

Solution
Question 2
1 / -0

The value of $${\int }_{0}^{1}\dfrac{dx}{x+\sqrt{1-x^{2}}}$$ is

A
$$\dfrac{\pi}{3}$$

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

C
$$\dfrac{1}{2}$$

D
$$\dfrac{\pi}{4}$$

Solution
Question 3
1 / -0

Let $$f: { (2,3)\rightarrow (0,1) }$$ be defined by $$f{ (x)=x-[x] }$$ then $$f^{ -1 }{ (x) }$$ equals

A
$$2x+1$$

B
$$x+1$$

C
$$x-1$$

D
$$x+2$$

Solution
Question 4
1 / -0

$$\displaystyle \int_{2}^{8}\dfrac {\sqrt {10-x}}{\sqrt {x}+\sqrt {10-x}}dx$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution
Question 5
1 / -0

$${\int}_{0}^{\pi}\dfrac{\sqrt{1-x}}{1+x}dx=$$

A
$$\dfrac{\pi}{2}$$

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

C
$$\dfrac{\pi}{2}+1$$

D
$$\pi+1$$

Solution
Question 6
1 / -0

$$\int _ { 0 } ^ { \pi / 4 } \tan ^ { 2 } x d x$$ equals -

A
$$\pi / 4$$

B
$$1 + ( \pi / 4 )$$

C
$$1 - ( \pi / 4 )$$

D
$$1 - ( \pi / 2 )$$

Solution
Question 7
1 / -0

The value of
$$\displaystyle\int _{ 5 }^{ 10 }{ \left( \sqrt { x+\sqrt { 20x-100 } } +\sqrt { x-\sqrt { 20x-100 } } \right) } dx$$ is

A
$$10\sqrt{5}$$

B
$$5\sqrt{5}$$

C
$$10\sqrt{2}$$

D
$$8\sqrt{2}$$

Solution
Question 8
1 / -0

Let $$f(x)= \left| \begin{matrix} \sec { x } & \cos { x } & \sec { ^{ 2 }x+\cot { x\csc { x } } } \\ \cos { ^{ 2 }x } & \cos { ^{ 2 }x } & \csc { ^{ 2 }x } \\ 1 & \cos { ^{ 2 }x } & \cos { ^{ 2 }x } \end{matrix} \right|$$. Then $${\int}_{0}^{\pi/2}f(x)dx$$ is equal to

A
$$\dfrac{15\pi}{60}$$

B
$$\dfrac{15\pi+32}{60}$$

C
$$\dfrac{15\pi-32}{60}$$

D
$$none\ of\ these$$
Question 9
1 / -0

Evaluate
$$\int \dfrac{dx}{\sqrt{1-x}}$$

A
$$\sin^{-1} \sqrt{x}$$

B
$$-\sin^{-1} \sqrt{x}+c$$

C
$$2\sqrt{1-x}+c$$

D
$$-2\sqrt{1-x}+c$$

Solution
Question 10
1 / -0

$$\displaystyle \overset{2\pi}{\underset{0}{\int}} x \,log \left(\dfrac{3 + \cos x}{3 - \cos x}\right)dx$$

A
$$\dfrac{\pi}{2} \,log \,3$$

B
$$\dfrac{\pi}{12} \,log \,3$$

C
$$\dfrac{\pi}{3} \,log \,3$$

D
$$0$$

Solution