CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 8 - MCQExams.com

Given, $$f(x) = \begin{vmatrix} 0 & {x^2 - \sin x} & {\cos x - 2} \\ {\sin x -x^2} & 0 & {1 - 2x} \\ {2 - \cos x} & {2x - 1} & 0 \end{vmatrix}$$, then $$\displaystyle \int f(x) dx$$ is equal to
  • $$\frac{x^3}{3} - x^2 \sin x + \sin 2x + C$$
  • $$\frac{x^3}{3} - x^2 \sin x + \cos 2x + C$$
  • $$\frac{x^3}{3} - x^2 \cos x + \cos 2x + C$$
  • None of the above
Let $$f(x)=\displaystyle \int_{-1}^{x}e^{t^2}dt$$ and $$h(x)=f(1+g(x))$$ where $$g(x)$$ is defined for all $$x, g'(x)$$ exists for all $$x,$$ and $$g(x) < 0$$ for $$x>0.$$ If $$h'(1)=e$$ and $$g'(1)=1,$$ then the possible values which $$g(1)$$ can take 
  • $$0$$
  • $$-1$$
  • $$-2$$
  • $$-4$$
If $$\displaystyle \int_{0}^{t} \dfrac{b x \cos 4 x-a \sin 4 x}{x^{2}} d x=\dfrac{a \sin 4 t}{t}-1,$$ where $$0<t<\dfrac{\pi}{4}$$then the values of $$a, b$$ are equal to
  • $$\dfrac{1}{4}, 1$$
  • $$-1,4$$
  • $$2,2$$
  • $$2,4$$
The value of $$\int_{a}^{b}(x-a)^{3}(b-x)^{4} d x$$ is
  • $$\dfrac{(b-a)^{4}}{6^{4}}$$
  • $$\dfrac{(b-a)^{8}}{280}$$
  • $$\dfrac{(b-a)^{7}}{7^{3}}$$
  • None of these
the value of $$I_{n+2}-I_n$$ is equal to 
  • $$n\pi$$
  • $$\pi$$
  • $$-\pi$$
  • $$0$$
The value of the definite integral 
$$\displaystyle \int_{0}^{\infty} \dfrac{dx}{(1+x^a)(1+x^2)}(a>0)$$ is
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{2}$$
  • $${\pi}$$
  • Some function of a
If $$\displaystyle I_{n}=\int_{0}^{1} \dfrac{d x}{\left(1+x^{2}\right)^{n}},$$ where $$n \in N,$$ which of the following statements hold good?
  • $$2 n I_{n+1}=2^{-n}+(2 n-1) I_{n}$$
  • $$I_{2}=\dfrac{\pi}{8}+\dfrac{1}{4}$$
  • $$I_{2}=\dfrac{\pi}{8}-\dfrac{1}{4}$$
  • $$I_{3}=\dfrac{3 \pi}{32}+\dfrac{1}{4}$$
Abscissae of point of intersection are in 
  • AP
  • GP
  • HP
  • None of these
The value of the definite integral $$\displaystyle \int_{0}^{\pi/2}sinx\space sin2x\space sin3xdx$$ is equal to  
  • $$\dfrac{1}{3}$$
  • $$-\dfrac{2}{3}$$
  • $$-\dfrac{1}{3}$$
  • $$\dfrac{1}{6}$$
The value of
$$\int_{2}^{5} \dfrac {\sqrt x}{\sqrt {x} + \sqrt {7-x}} dx$$
is :
  • 3
  • 2
  • $$\dfrac {3}{2}$$
  • $$\dfrac {1}{2}$$
If $$A(x) = \int_{0}^{x} \theta ^2 d\theta$$
then value of $$A(3)$$ will be :
  • 9
  • 27
  • 3
  • 81
$$\int_{a-c}^{b-c} f(x+c) dx $$ is:
  • $$\int_{a}^{b} f(x+c) dx $$
  • $$\int_{a}^{b} f(x) dx $$
  • $$\int_{a-2c}^{b-2c} f(x) dx $$
  • $$\int_{a}^{b} f(x+2c) dx $$
$$\int_{0}^{log 5} \displaystyle \frac{e^{x}\sqrt{e^{x}-1}}{e^{x}+3}  dx =$$
  • $$3+ 2 \pi$$
  • $$4 - \pi$$
  • $$2 + \pi$$
  • $$\pi -4$$
If  $$\displaystyle I_n=\int_{0}^{\tfrac{\pi}{4}} \tan^nx\sec^2xdx,$$ then  $$I_1,   I_2,  I_3..$$ are  in
  • A.P.
  • G.P.
  • H.P.
  • A.G.P.
Evaluate the integral
$$\displaystyle \int_{0}^{\pi/4}\frac{ {s}i {n}\theta+ {c} {o} {s}\theta}{9+16 {s}i {n}2\theta} \ {d}\theta $$
  • $$\displaystyle \frac{1}{20} \log 2$$ 
  • $$\displaystyle \frac{1}{20} \log 3$$ 
  • $$\log 3$$
  • $$\log 2$$
The value of the definite integral $$\displaystyle \int_{ 0 }^{\sqrt{{\ln (\displaystyle \frac{\pi}{2})}}}\cos(e^{x^{2}})2xe^{x^{2}} \:dx$$ is:
  • $$1$$
  • $$1+ (\sin1)$$
  • $$1- (\sin1)$$
  • $$(\sin1)-1$$
Evaluate: $$\displaystyle \int_{0}^{\pi}\frac{dx}{5+4\cos x}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{3}$$
  • $$-\pi$$
$$\int ^{\pi /2}_{-\pi /2} cost . \sin(2t-\displaystyle \frac{\pi}{4})dt=$$
  • $$\displaystyle \frac{\sqrt{2}}{3}$$
  • $$-\displaystyle \frac{\sqrt{2}}{3}$$
  • $$\displaystyle \frac{\sqrt{3}}{1}$$
  • $$\displaystyle \frac{1}{\sqrt{3}}$$

$$\displaystyle \int_{0}^{a}\frac{dx}{x+\sqrt{a^{2}-x^{2}}}=$$
  • $$\pi$$
  • $$\dfrac{\pi}{3}$$
  • $$-\pi$$
  • $$\dfrac{\pi}{4}$$

$$\displaystyle \int_{0}^{\pi/2}\sqrt{\cos x}\sin^{5}xdx=$$
  • $$\displaystyle \frac{34}{231}$$
  • $$\displaystyle \frac{64}{231}$$
  • $$\displaystyle \frac{30}{321}$$
  • $$\displaystyle \frac{128}{231}$$
$$\displaystyle \int_{1}^{2}\mathrm{x}^{2\mathrm{x}}[1+\log \mathrm{x}]\mathrm{d}\mathrm{x}=$$
  • $$\displaystyle \frac{9}{2}$$
  • $$\displaystyle \frac{11}{2}$$
  • $$\displaystyle \frac{13}{2}$$
  • $$\displaystyle \frac{15}{2}$$
If $$\mathrm{a}>\mathrm{b}$$ then $$\displaystyle \int_{0}^{\pi}\frac{\mathrm{d}\mathrm{x}}{\mathrm{a}+\mathrm{b}\sin \mathrm{x}}=$$
  • $$\displaystyle \frac{2}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\cot^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$$
  • $$\displaystyle \frac{1}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\cot^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$$
  • $$\displaystyle \frac{1}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\tan^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$$
  • $$\displaystyle \frac{2}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\tan^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$$
lf $$\displaystyle \int_{0}^{\infty}e^{-\mathrm{x}^{2}}\mathrm{d}\mathrm{x}=\frac{\sqrt{\pi}}{2}$$, then $$\displaystyle \int_{0}^{\infty}e^{-ax^{2}}dx,\ \mathrm{a}>0$$ is
  • $$\displaystyle \frac{\sqrt{\pi}}{2}$$
  • $$\displaystyle \frac{\sqrt{\pi}}{2a}$$
  • $$2 \displaystyle \frac{\sqrt{\pi}}{a}$$
  • $$\displaystyle \frac{1}{2}\sqrt{\frac{\pi}{a}}$$
Let $$\displaystyle\frac{d}{dx}(F(x))=\frac{e^{\displaystyle\sin{x}}}{x}$$, $$x>0$$. If $$\displaystyle\int_1^4{\frac{2e^{\displaystyle\sin{x^2}}}{x}dx}=F(k)-F(1)$$, then the possible value of $$k$$ is
  • 10
  • 14
  • 16
  • 18
Observe the following Lists
List-IList-II
A: $$\displaystyle \int_{-2}^{2}\frac{1}{4+x^{2}}dx$$1) $$\displaystyle \frac{\pi}{3}$$
B: $$\displaystyle \int_{1}^{2}\frac{1}{x\sqrt{x^{2}-1}}dx$$2) 0
C: $$\displaystyle \int_{0}^{\pi}\cos 3x.\cos 2xdx$$3) $$\displaystyle \frac{\pi}{4}$$
4) $$\displaystyle \frac{\pi}{2}$$
  • A-3, B-1, C-4
  • A-3, B-1, C-2
  • A-1, B-3, C-2
  • A-4, B-1, C-2
If $$\displaystyle \int_{0}^{\pi/3}\frac{\cos x}{3+4\sin x}dx=k\log(\frac{3+2\sqrt{3}}{3})$$, then $$\mathrm{k}$$ is equal to
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{8}$$
The value of the integral 
$$\displaystyle \int_{0}^{3\alpha}\text{cosec}(x -\alpha)\text{cosec}(x-2\alpha) dx$$ is
  • $$2\displaystyle \sec\alpha\log(\frac{1}{2}\text{cosec}\alpha)$$
  • $$2\displaystyle \sec\alpha\log(\frac{1}{2}\sec\alpha)$$
  • $$2 \text{cosec}\alpha\log(\sec\alpha)$$
  • $$2 \displaystyle \text{cosec}\alpha\log(\frac{1}{2}\sec\alpha)$$
If $$I_{n}=\displaystyle \int_{0}^{\frac{\pi}{4}}\tan^{n} dx,$$ then $$\dfrac{1}{I_{2}+I_{4}},\dfrac{1}{I_{3}+I_{5}},\dfrac{1}{I_{4}+I_{6}}\cdots$$ form
  • an A.P
  • a G.P
  • a H.P
  • an AGP
If $$f(x) = x - x^2 +1$$ & $$g(x)=max\left \{ f(t) ;0\leq t< x \right \}$$, then $$\overset {1}{\underset { 0 }{ \int } }  g (x) dx = ?$$
  • $$\dfrac{29}{24}$$
  • $$\dfrac{7}{6}$$
  • $$\dfrac{5}{4}$$
  • none of these
$$\displaystyle \int_{0}^{1}\frac{3^{x+1}-4^{x-1}}{12^{x}}dx=$$
  • $$\displaystyle \frac{9}{4}log_{4}\, e$$
  • $$ \displaystyle \frac{9}{4}log_{4}\, e-\frac{1}{6}log_{3}e$$
  • $$ \displaystyle log_{4}3$$
  • None of these
Let $$\displaystyle \frac{d}{dx}F(x)=\frac{e^{{s}{m}{x}}}{x}$$ , $$x>0$$. lf $$\displaystyle \int_{1}^{4}\frac{3}{x}e^{{s}{m}{x}^{3}}dx=F(k)-F(1)$$, then one of the possible values of $${k}$$ is
  • $$16$$
  • $$62$$
  • $$64$$
  • $$15$$
$$\displaystyle\int_0^\infty{f\left(x+\frac{1}{x}\right).\frac{\ln{x}}{x}dx}$$ is equal to
  • 0
  • 1
  • $$\displaystyle\frac{1}{2}$$
  • cannot be evaluated
$$\displaystyle \int_{0}^{1}\displaystyle \frac{x(2x^{2}+1)}{x^{8}+2x^{6}-x^{2}+1}dx=$$
  • $$\displaystyle \frac{\pi }{3}$$
  • $$\displaystyle \frac{\pi }{2\sqrt{3}}$$
  • $$\displaystyle \frac{2\pi }{3}$$
  • $$\displaystyle \frac{\pi }{6}$$
If $$I_{n}=\displaystyle \int_{0}^{\infty}e^{-x}x^{n-1} dx$$, then $$\displaystyle \int_{0}^{\infty}e^{-\lambda x}x^{n-1} dx$$ is equal to?
  • $$\lambda I_{\mathrm{n}}$$
  • $$\displaystyle \frac{1}{\lambda}I_{\mathrm{n}}$$
  • $$\displaystyle \frac{I_{\mathrm{n}}}{\lambda^{\mathrm{n}}}$$
  • $$\lambda^{n}I_{n}$$
The value of $$\displaystyle \int_{0}^{\pi}\dfrac {dx}{1+2sin^2x}$$ is
  • $$0$$
  • $$\dfrac {\pi}{2\sqrt 3}$$
  • $$\dfrac {\pi}{\sqrt 3}$$
  • None of these
The value of the integral $$\displaystyle \int_{0 }^{\log5}\frac{e^x\sqrt{e^x-1}}{e^x+3}\:dx$$ is
  • $$3+2\pi$$
  • $$4-\pi$$
  • $$2+\pi$$
  • none of these
The value of the integral  $$\displaystyle \int_{0}^{\frac{1}{\sqrt{3}}}\frac{dx}{(1+x^2)\sqrt{1-x^2}}$$
  • $$\displaystyle \frac{\pi }{2\sqrt{2}}$$
  • $$\displaystyle \frac{\pi }{4\sqrt{2}}$$
  • $$\displaystyle \frac{\pi }{8\sqrt{2}}$$
  • none of these
The value of $$\displaystyle {\int}_0^{\pi}\frac{dx}{1+2 \sin^2 x}$$ is
  • $$0$$
  • $$\displaystyle \frac{\pi}{2\sqrt{3}}$$
  • $$\displaystyle \frac{\pi}{\sqrt{3}}$$
  • none of these
$$\displaystyle \int_{- \dfrac{\pi}{2}}^{\dfrac{\pi}{2}}\sin^{2}x. \cos^{3} x dx$$ is equal to
  • $$\displaystyle \dfrac{4\pi}{30}$$
  • $$\displaystyle \dfrac{4}{15}$$
  • $$\displaystyle \dfrac{\pi}{30}$$
  • $$\displaystyle \dfrac{1}{15}$$
$$\displaystyle \int_{\frac{5}{2}}^{5}\frac{\sqrt{(25-x^2)^3}}{x^4}\:dx$$ is equal to
  • $$\displaystyle \frac{\pi }{6}$$
  • $$\displaystyle \frac{2\pi }{3}$$
  • $$\displaystyle \frac{5\pi }{6}$$
  • $$\displaystyle \frac{\pi }{3}$$
If $$\displaystyle \int_{\log 2}^{x}\displaystyle \frac{dx}{\sqrt{e^{x}-1}}= \displaystyle \frac{\pi }{6}$$, the value of $$x$$ is
  • $$4$$
  • $$\log 8$$
  • $$\log 4$$
  • none of these
The minimum value of the function f(x) = $$\int^x_0 \frac{d \theta}{cos \theta} + \int^{\pi/2}_x \frac{d \theta}{sin \theta} $$ where $$x \in [0, \frac{\pi}{2}], $$ is  
  • $$2ln(\sqrt{2} + 1)$$
  • $$ln(2\sqrt{2} + 2)$$
  • $$ln(\sqrt{3} + 2)$$
  • $$ln(\sqrt{2} + 3)$$
The value of $$\int _{-\pi /2}^{\pi /2}\left (psin^3 x+qsin^4 x +r sin^5 x  \right )$$ does not depend on
  • p, q, r
  • p, r only
  • p only
  • q, r only
If $$ \displaystyle \int_{0}^{1} \frac{\tan^{-1} x}{x}dx$$  is equal to
  • $$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sin x}{x}dx$$
  • $$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{x}{\sin x}dx$$
  • $$\displaystyle \frac{1}{2} \int_{0}^{\frac{\pi}{2}}\frac{\sin x}{x}dx$$
  • $$\displaystyle \frac{1}{2} \int_{0}^{\frac{\pi}{2}}\frac{x}{\sin x}dx$$
Let  $$ \displaystyle I_1=\int_{0}^{1}\frac{e^x dx}{1+x}$$ and $$\displaystyle I_2=\int_{0}^{1}\dfrac{x^2 dx}{e^{x^3}(2-x^3)}$$.
Then $$\dfrac{I_1}{I_2}$$ is equal to?
  • $$\displaystyle \dfrac{3}{e}$$
  • $$\displaystyle \frac{e}{3}$$
  • $$\displaystyle {3}{e}$$
  • $$\displaystyle \dfrac{1}{3e}$$
The value of $$\displaystyle \int_{0}^{\infty} \frac {dx}{1 + x^4}$$ is
  • same as that of $$\displaystyle \int_{0}^{\infty} \frac {x^2 + 1dx}{1 + x^4}$$
  • $$\displaystyle \frac {\pi}{2\sqrt{2}}$$
  • same as that of $$\displaystyle \int_{0}^{\infty} \frac {x^2 \: dx}{1 + x^4}$$
  • $$\displaystyle \frac {\pi}{\sqrt{2}}$$
The value of the integral $$\displaystyle \int_{-\pi}^{\pi} (\cos  ax - \sin  b  x)^2 dx$$, where $$a$$ and $$b$$ are integers, is
  • $$2\pi (1+a+b)$$
  • $$0$$
  • $$\pi$$
  • $$2 \pi$$
Evaluate $$\displaystyle\int_0^{\displaystyle\frac{\pi}{4}}{\frac{\sin{x}+\cos{x}}{9+16\{1-{(\sin{x}-\cos{x})}^2\}}dx}$$
  • $$\displaystyle\frac{1}{20}(\log{\sqrt{3}})$$
  • $$\displaystyle\frac{1}{10}(\log{3})$$
  • $$\displaystyle\frac{1}{20}(\log{3})$$
  • $$\displaystyle\frac{1}{20}(\log{3}-\log{2})$$
If $$\displaystyle \int_{1}^{2} e^{x^2} dx= a$$, then $$\displaystyle \int_{e}^{e^4}\sqrt{\ln x} \:dx$$ is equal to
  • $$2e^4-2e-a$$
  • $$2e^4-e-a$$
  • $$2e^4-e-2a$$
  • $$e^4-e-a$$
$$\displaystyle \int_{ 0 }^{\infty}\frac{ \:dx}{\left [ x+\sqrt{x^2+1} \right ]^3}$$ is equal to
  • $$\displaystyle \frac{3}{8}$$
  • $$\displaystyle \frac{1}{8}$$
  • $$\displaystyle -\frac{3}{8}$$
  • none of these
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