CBSE Questions for Class 12 Commerce Maths Integrals Quiz 7 - MCQExams.com

Evaluate $$\displaystyle\int^4_1x\sqrt{x}dx$$
  • $$12.8$$
  • $$12.4$$
  • $$7$$
  • None of these
Evaluate $$\displaystyle\int^{\pi/4}_0\tan^2xdx$$
  • $$\left(1-\dfrac{\pi}{4}\right)$$
  • $$\left(1+\dfrac{\pi}{4}\right)$$
  • $$\left(1-\dfrac{\pi}{2}\right)$$
  • $$\left(1+\dfrac{\pi}{2}\right)$$
Evaluate : $$\displaystyle\int^1_0\dfrac{dx}{\sqrt{5x+3}}$$
  • $$\dfrac{2}{5}(\sqrt{8}-\sqrt{3})$$
  • $$\dfrac{2}{5}(\sqrt{8}+\sqrt{3})$$
  • $$\dfrac{2}{5}\sqrt{8}$$
  • None of these
Evaluate : $$\displaystyle\int^2_0\sqrt{6x+4}dx$$
  • $$\dfrac{64}{9}$$
  • $$7$$
  • $$\dfrac{56}{9}$$
  • $$\dfrac{60}{9}$$
Evaluate $$\displaystyle\int^2_0\dfrac{dx}{\sqrt{4-x^2}}$$
  • $$1$$
  • $$\sin^{-1}\dfrac{1}{2}$$
  • $$\dfrac{\pi}{4}$$
  • None of these
Evaluate $$\displaystyle\int^1_0\dfrac{x^3}{(1+x^8)}dx$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{8}$$
  • $$\dfrac{\pi}{16}$$
Evaluate $$\displaystyle\int^{\pi/2}_{\pi/3} cosec xdx$$
  • $$\dfrac{1}{2} log 2$$
  • $$\dfrac{1}{2} log 3$$
  • $$-log 2$$
  • None of these
Evaluate $$\displaystyle\int^{\pi/2}_{\pi/4}\cot xdx$$
  • $$log 2$$
  • $$2 log 2$$
  • $$\dfrac{1}{2}log 2$$
  • None of these
Evaluate $$\displaystyle\int^{\pi/2}_0\dfrac{\cos x}{(1+\sin^2x)}dx$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
  • $$\pi$$
  • None of these
Evaluate $$\displaystyle\int^{\sqrt{8}}_{\sqrt{3}}x\sqrt{1+x^2}dx$$
  • $$\dfrac{19}{3}$$
  • $$\dfrac{19}{6}$$
  • $$\dfrac{38}{3}$$
  • $$\dfrac{9}{4}$$
Evaluate $$\displaystyle\int^{\pi/2}_0\cos^3xdx $$
  • $$1$$
  • $$\dfrac{3}{4}$$
  • $$\dfrac{2}{3}$$
  • None of these
Evaluate : $$\displaystyle\int^1_0\sqrt{\dfrac{1-x}{1+x}}dx$$
  • $$\dfrac{\pi}{2}$$
  • $$\left(\dfrac{\pi}{2}-1\right)$$
  • $$\left(\dfrac{\pi}{2}+1\right)$$
  • None of these
Evaluate : $$\displaystyle\int^1_0\dfrac{dx}{(1+x+x^2)}$$
  • $$\dfrac{\pi}{\sqrt{3}}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{3\sqrt{3}}$$
  • None of these
Evaluate : $$\displaystyle\int^a_{-a}\sqrt{\dfrac{a-x}{a+x}}dx$$
  • $$a\pi$$
  • $$\dfrac{a\pi}{2}$$
  • $$2a\pi$$
  • None of these
Evaluate$$\displaystyle\int^{\pi/2}_0e^x\left(\dfrac{1+\sin x}{1+\cos x}\right)dx$$
  • $$0$$
  • $$\dfrac{\pi}{4}$$
  • $$e^{\pi/2}$$
  • $$(e^{\pi/2}-1)$$
Evaluate $$\displaystyle\int^{\pi}_0\dfrac{dx}{(1+\sin x)}$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$2$$
  • $$0$$
Evaluate : $$\displaystyle\int^{\sqrt{2}}_0\sqrt{2-x^2}dx$$
  • $$\pi$$
  • $$2\pi$$
  • $$\dfrac{\pi}{2}$$
  • None of these
Evaluate : $$\displaystyle\int^1_0\dfrac{(1-x)}{(1+x)}dx$$
  • $$(\log 2+1)$$
  • $$(\log 2-1)$$
  • $$(2 \log 2-1)$$
  • $$(2 \log 2+1)$$
Evaluate : $$\displaystyle\int^9_0\dfrac{dx}{(1+\sqrt{x})}$$
  • $$(3-2 log 2)$$
  • $$(3+2 log 2)$$
  • $$(6-2 log 4)$$
  • $$(6+2 log 4)$$
Evaluate $$\displaystyle\int^1_0\dfrac{xe^x}{(1+x)^2}dx$$
  • $$\left(\dfrac{e}{2}-1\right)$$
  • $$(e-1)$$
  • $$e(e-1)$$
  • None of these
Evaluate $$\displaystyle\int^{1}_0\dfrac{(1-x)}{(1+x)}dx$$
  • $$\dfrac{1}{2}log 2$$
  • $$(2log 2+1)$$
  • $$(2log 2-1)$$
  • $$\left(\dfrac{1}{2}log 2-1\right)$$
The value of $$\displaystyle\int^{199\pi/2}_{-\pi/2}\sqrt{(1+\cos 2x)}dx$$ is?
  • $$50\sqrt{2}$$
  • $$100\sqrt{2}$$
  • $$150\sqrt{2}$$
  • $$200\sqrt{2}$$
Evaluate : $$\displaystyle\int^2_1|x^2-3x+2|dx$$
  • $$\dfrac{-1}{6}$$
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{2}{3}$$
$$\displaystyle\int^a_{-a}x|x|dx=?$$
  • $$0$$
  • $$2a$$
  • $$\dfrac{2a^3}{3}$$
  • None of these
$$\displaystyle\int^1_{-2}\dfrac{|x|}{2}dx=?$$
  • $$3$$
  • $$2.5$$
  • $$1.5$$
  • None of these
Let $$\displaystyle\dfrac{x^{1/2}}{\sqrt{1-x^{3}}}dx=\dfrac{2}{3}gof(x)+c$$ then
  • $$f(x)=\sqrt{x}$$
  • $$f(x)=x^{3/2}$$
  • $$f(x)=x^{2/3}$$
  • $$g(x)=\sin^{-1}x$$
Evaluate : $$\displaystyle\int^1_0|2x-1|dx$$
  • $$2$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$0$$
Evaluate : $$\displaystyle\int^2_{-2}|x|dx$$
  • $$4$$
  • $$3.5$$
  • $$2$$
  • $$0$$
The value of $$\int_{1}^{e} \dfrac{1+x^{2} \ln x}{x+x^{2} \ln x} d x$$ is
  • $$e$$
  • $$\ln (1+e)$$
  • $$e+\ln (1+e)$$
  • $$e-\ln (1+e)$$
$$I_{1}=\int_{0}^{\frac{\pi}{2}} \dfrac{\sin x-\cos x}{1+\sin x \cos x} d x, I_{2}=\int_{0}^{2 \pi} \cos ^{6} x d x$$$$I_{3}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{3} x d x, I_{4}=\int_{0}^{1} \ln \left(\dfrac{1}{x}-1\right) d x,$$ then
  • $$I_{2}=I_{3}=I_{4}=0, I_{1} \neq 0$$
  • $$I_{1}=I_{2}=I_{3}=0, I_{4} \neq 0$$
  • $$I_{1}=I_{3}=I_{4}=0, I_{2} \neq 0$$
  • $$I_{1}=I_{2}=I_{3}=0, I_{4} \neq 0$$
Evaluate : $$\displaystyle\int^1_{-2}|2x+1|dx$$
  • $$\dfrac{5}{2}$$
  • $$\dfrac{7}{2}$$
  • $$\dfrac{9}{2}$$
  • $$4$$
Let $$f : R \rightarrow R$$ be a function as $$f(x) = (x - 1)(x + 2)(x - 3)(x - 6) - 100$$. If $$g(x)$$ is a polynomial of degree $$\leq 3$$ such that $$\displaystyle \int \frac{g(x)}{f(x)} dx$$ does not contain any logarithm function and $$g(-2) = 10$$. Then

$$\displaystyle \int \frac{g(x)}{f(x)} dx$$, equals
  • $$\tan^{-1} \left ( \frac{x - 2}{2}
    \right ) + c$$
  • $$\tan^{-1} \left ( \frac{x - 1}{1}
    \right ) + c$$
  • $$\tan^{-1} (x) + c$$
  • None of these
The value of the definite integral $$\displaystyle \int_{0}^{\pi/2}sinx\space sin2x\space sin3xdx$$ is equal to  
  • $$\dfrac{1}{3}$$
  • $$-\dfrac{2}{3}$$
  • $$-\dfrac{1}{3}$$
  • $$\dfrac{1}{6}$$
If $$\displaystyle \int f(x) dx = F(x)$$, then $$\displaystyle \int x^3 f(x^2) dx$$ is equal to
  • $$\frac{1}{2} [x^2$$ {$$F(x)$$}$$^2 - \displaystyle \int$$ {$$F(x)$$}$$^2 dx$$]
  • $$\frac{1}{2} [x^2 F(x^2) - \displaystyle \int F(x^2) d (x^2)]$$
  • $$\frac{1}{2} [x^2 F(x^2) - \frac{1}{2} \displaystyle \int$$ {$$F(x)$$}$$^2 dx$$]
  • None of the above
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
$$\int_{0}^{log 5} \displaystyle \frac{e^{x}\sqrt{e^{x}-1}}{e^{x}+3}  dx =$$
  • $$3+ 2 \pi$$
  • $$4 - \pi$$
  • $$2 + \pi$$
  • $$\pi -4$$
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
$$\displaystyle \int_{0}^{\pi/4}\frac{ {s}i {n}\theta+ {c} {o} {s}\theta}{9+16 {s}i {n}2\theta} \ {d}\theta $$
  • $$\displaystyle \frac{1}{20} \log 2$$ 
  • $$\displaystyle \frac{1}{20} \log 3$$ 
  • $$\log 3$$
  • $$\log 2$$
Evaluate: $$\displaystyle \int_{0}^{\pi}\frac{dx}{5+4\cos x}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{3}$$
  • $$-\pi$$

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

$$\displaystyle \int_{0}^{\pi/2}\sqrt{\cos x}\sin^{5}xdx=$$
  • $$\displaystyle \frac{34}{231}$$
  • $$\displaystyle \frac{64}{231}$$
  • $$\displaystyle \frac{30}{321}$$
  • $$\displaystyle \frac{128}{231}$$
$$\displaystyle \int_{0}^{\pi/2}\frac{1}{a+bcosx}dx=$$, where $$a>|b|$$
  • $$\displaystyle \frac{2}{\sqrt{a^{2}-b^{2}}}\tan^{{-1}}\sqrt{\frac{a+b}{a-b}}$$
  • $$\displaystyle \frac{2}{\sqrt{a^{2}-b^{2}}}cot^{-l} \sqrt{\frac{a-b}{a+b}}$$
  • $$\displaystyle \frac{2}{\sqrt{a^{2}-b^{2}}}\tan^{{-1}}\sqrt{\frac{a-b}{a+b}}$$
  • $$\displaystyle \frac{\pi}{\sqrt{a^{2}-b^{2}}}$$
$$\displaystyle \int_{1}^{2}\mathrm{x}^{2\mathrm{x}}[1+\log \mathrm{x}]\mathrm{d}\mathrm{x}=$$
  • $$\displaystyle \frac{9}{2}$$
  • $$\displaystyle \frac{11}{2}$$
  • $$\displaystyle \frac{13}{2}$$
  • $$\displaystyle \frac{15}{2}$$
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
  • $$\displaystyle \frac{\sqrt{\pi}}{2}$$
  • $$\displaystyle \frac{\sqrt{\pi}}{2a}$$
  • $$2 \displaystyle \frac{\sqrt{\pi}}{a}$$
  • $$\displaystyle \frac{1}{2}\sqrt{\frac{\pi}{a}}$$
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
$$\displaystyle \int_{0}^{1}\frac{3^{x+1}-4^{x-1}}{12^{x}}dx=$$
  • $$\displaystyle \frac{9}{4}log_{4}\, e$$
  • $$ \displaystyle \frac{9}{4}log_{4}\, e-\frac{1}{6}log_{3}e$$
  • $$ \displaystyle log_{4}3$$
  • None of these
Let f be a function defined for every x, such that f'' = -f ,f(0)=0, f' (0) = 1, then f(x) is equal to
  • tanx
  • $$e^{x}-1$$
  • sinx
  • 2sinx
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
  • $$16$$
  • $$62$$
  • $$64$$
  • $$15$$
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