Integrals Test

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Integrals Test - 60

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Weekly Quiz Competition

Question 1
1 / -0

$$\displaystyle \int_{0}^{\pi/4}{\dfrac{x.\sin x}{\cos^{3}x}dx}$$ equals to :

A
$$\dfrac{\pi}{4}+\dfrac{1}{2}$$

B
$$\dfrac{\pi}{4}-\dfrac{1}{2}$$

C
$$\dfrac{\pi}{4}$$

D
$$\dfrac{\pi}{4}+1$$

Solution

$$\displaystyle\int_{0}^{\frac{\pi}{4}}{\dfrac{x\sin{x}}{{\cos}^{3}{x}}dx}$$

$$=\displaystyle\int_{0}^{\frac{\pi}{4}}{x\tan{x}{\sec}^{2}{x}dx}$$

Let $$t=\tan{x}\Rightarrow\,dt={\sec}^{2}{x}dx$$

$$\Rightarrow\,x={\tan}^{-1}{t}$$

When $$x=0\Rightarrow\,t=0$$

When $$x=\dfrac{\pi}{4}\Rightarrow\,t=1$$

$$=\displaystyle\int_{0}^{1}{{\tan}^{-1}{t}\,dt}$$

Let $$u={\tan}^{-1}{t}\Rightarrow\,du=\dfrac{1}{1+{t}^{2}}dt$$

$$dv=t\Rightarrow\,v=\dfrac{{t}^{2}}{2}$$

$$=\left[{\tan}^{-1}{t}\dfrac{{t}^{2}}{2}\right]_{0}^{1}-\displaystyle\int_{0}^{1}{\dfrac{{t}^{2}}{2}\times\dfrac{1}{1+{t}^{2}}dt}$$

$$=\dfrac{1}{2}\left[{t}^{2}{\tan}^{-1}{t}\right]_{0}^{1}-\dfrac{1}{2}\displaystyle\int_{0}^{1}{\dfrac{{t}^{2}}{1+{t}^{2}}dt}$$

$$=\dfrac{1}{2}\left[{t}^{2}{\tan}^{-1}{t}\right]_{0}^{1}-\dfrac{1}{2}\displaystyle\int_{0}^{1}{\dfrac{{t}^{2}+1-1}{1+{t}^{2}}dt}$$

$$=\dfrac{1}{2}\left[{t}^{2}{\tan}^{-1}{t}\right]_{0}^{1}-\dfrac{1}{2}\displaystyle\int_{0}^{1}{\dfrac{{t}^{2}+1}{1+{t}^{2}}dt}+\dfrac{1}{2}\displaystyle\int_{0}^{1}{\dfrac{dt}{1+{t}^{2}}}$$

$$=\dfrac{1}{2}\left[{\tan}^{-1}{1}-0\right]-\dfrac{1}{2}\left[t\right]_{0}^{1}+\dfrac{1}{2}\left[{\tan}^{-1}{t}\right]_{0}^{1}$$

$$=\dfrac{1}{2}\left[\dfrac{\pi}{4}-0\right]-\dfrac{1}{2}\left[1-0\right]_{0}^{1}+\dfrac{1}{2}\left[\dfrac{\pi}{4}-0\right]_{0}^{1}$$

$$=\dfrac{\pi}{8}-\dfrac{1}{2}+\dfrac{\pi}{8}$$

$$=\dfrac{2\pi}{8}-\dfrac{1}{2}$$

$$=\dfrac{\pi}{4}-\dfrac{1}{2}$$
Question 2
1 / -0

The value of $$\cfrac { \int _{ 0 }^{ \pi /2 }{ { \left( \sin { x } \right) }^{ \sqrt { 3 } +1 } } dx }{ \int _{ 0 }^{ \pi /2 }{ { \left( \sin { x } \right) }^{ \sqrt { 3 } -1 } } } $$ is

A
$$\cfrac { \sqrt { 3 } +1 }{ \sqrt { 3 } -1 } $$

B
$$\cfrac { \sqrt { 3 } -1 }{ \sqrt { 3 } +1 } $$

C
$$\cfrac { \sqrt { 3 } +1 }{ \sqrt { 3 } } $$

D
none of these

Solution
Question 3
1 / -0

The value of $$\displaystyle \int _{-1}^{1} \log{\left(\dfrac{2-x}{2+x}\right)}\sin^{2}{x}dx$$

A
$$1$$

B
$$-1$$

C
$$2$$

D
$$0$$

Solution
Question 4
1 / -0

$$\displaystyle\int^1_0\tan^{-1}\left[\dfrac{2x-1}{1+x-x^2}\right]dx=?$$

A
$$0$$

B
$$1/2$$

C
$$1$$

D
$$\pi/6$$

Solution
Question 5
1 / -0

$$\displaystyle\int _{ -1 }^{ 1 }{ x\ell n\left( 1+{ e }^{ x } \right) dx } =$$

A
$$0$$

B
$$\ell n(1+e)$$

C
$$\ell n(1+e)-1$$

D
$$1/3$$

Solution
Question 6
1 / -0

$$\displaystyle\int^{2+\sqrt{3}}_{2-\sqrt{3}}\dfrac{xdx}{(1+x)(1+x^2)}=?$$

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

B
$$\dfrac{\pi}{6}$$

C
$$\dfrac{\pi}{12}$$

D
$$\dfrac{\pi}{24}$$

Solution
Question 7
1 / -0

$$\displaystyle\int^{2\pi}_0[\sin x]dx$$.

A
0

B
$$-\pi$$

C
$$2\pi$$

D
$$-2\pi$$

Solution
Question 8
1 / -0

$$\int _{ \pi /4 }^{ 3\pi /4 }{ \dfrac { dx }{ 1+\cos { x } } }$$ is equal to

A
$$-2$$

B
$$2$$

C
$$4$$

D
$$-1$$

Solution
Question 9
1 / -0

Evaluate: $$\displaystyle \int \dfrac { 1 } { x ^ { 2 } \left( x ^ { 4 } + 1 \right) ^ { \frac { 3 } { 4 } } } d x ; x = 0$$

A
$$\dfrac { \left( x ^ { 4 } - 1 \right) ^ { \frac { 1 } { 4 } } } { x } + c$$

B
$$-\dfrac { \left( x ^ { 4 } + 1 \right) ^ { \frac { 1 } { 4 } } } { x } + c$$

C
$$\dfrac { \sqrt { x ^ { 4 } + 1 } } { x } + c$$

D
None of these

Solution

$$I=\displaystyle\int \frac{dx}{x^{2}(1+x^{4})^{3/4}}$$

$$=\displaystyle\int \frac{dx}{x^{2}\dfrac{(1+x^{4})^{3/4}.x^3}{x^{3}}}$$

$$I=\displaystyle\int \frac{dx}{x^{5}(\dfrac{(1+x^{4})^{1/4}}{x})^{3}}$$

$$u=\dfrac{(1+x^{4})^{1/4}}{x}$$

$$(ux)^{4}= x^{4}+1$$

$$u^{4}= 1+\dfrac{1}{x^{4}}$$

$$4u^{3}du = -\dfrac{4\, dx}{x^{5}}$$

$$u^{3}du=-\dfrac{dx}{x^{5}}$$

$$I=\displaystyle\int \dfrac{-u^{3}du}{u^{3}}=-\int du$$

$$= -u+c$$

$$=-\dfrac{(1+x^{4})^{1/4}}{x}+c$$
Question 10
1 / -0

If $$\displaystyle \int _{0}^{1}\cot^{-1}(1+x^{2}-x)dx=k\left(\dfrac {\pi}{4}-\log_{e}\sqrt {2}\right)$$, then the value of $$k$$ is equal to

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$2$$

Solution