# CBSEtips.in

## Thursday 28 January 2021

### CBSE Class 12 Maths - MCQ and Online Tests - Unit 7 - Integrals

#### CBSE Class 12 Maths – MCQ and Online Tests – Unit 7 – Integrals

Every year CBSE conducts board exams for 12th standard. These exams are very competitive to all the students. So our website provides online tests for all the 12th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

These online tests are based on latest CBSE Class 12 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.

#### CBSE Class 12 Maths – MCQ and Online Tests – Unit 7 – Integrals

Question 1.
∫$$\frac{(1+logx)^2}{1+x^2}$$ dx =
(a) $$\frac{1}{3}$$(1+log)³ + c
(b) $$\frac{1}{2}$$(1+log)² + c
(c) log (log 1 + x) + 2
(d) None of these

Answer: (a) $$\frac{1}{3}$$(1+log)³ + c

Question 2.
$$\int_{0}^{1}\frac{(tan^{-1}x)^2}{1+x^2}$$ dx =
(a) 1
(b) $$\frac{π^2}{64}$$
(c) $$\frac{π^2}{192}$$
(d) None of these

Answer: (c) $$\frac{π^2}{192}$$

Question 3.
$$\int_{-2}^{2}$$ |x|dx =
(a) 0
(b) 2
(c) 1
(d) 4

Question 4.
If x > a, ∫$$\frac{dx}{x^2-a^2}$$ =
(a) $$\frac{2}{2a}$$ log $$\frac{x-a}{x+a}$$ + k
(b) $$\frac{2}{2a}$$ log $$\frac{x+a}{x-a}$$ + k
(c) $$\frac{1}{a}$$ log(x² – a²) + k
(d) log(x + $$\sqrt{x^2-a^2}$$ + k)

Answer: (a) $$\frac{2}{2a}$$ log $$\frac{x-a}{x+a}$$ + k

Question 5.
$$\int \frac{x+\sin x}{1+\cos x}$$ dx is equal to
(a) log |1 + cos x | + c
(b) log | x + sin x | + c
(c) x – tan + c
(d) x. tan $$\frac{x}{2}$$ + c

Answer: (d) x. tan $$\frac{x}{2}$$ + c

Question 6.
∫$$\frac{x^4+1}{x^2+1}$$ dx is equal to
(a) $$\frac{x^3}{3}$$ + x + tan-1 x + c
(b) $$\frac{x^3}{3}$$ – x + tan x + c
(c) $$\frac{x^3}{3}$$ + x + 2tan-1 x + c
(d) $$\frac{x^3}{3}$$ – x + 2tan-1 x + c

Answer: (d) $$\frac{x^3}{3}$$ – x + 2tan-1 x + c

Question 7.
∫1.dx =
(a) x + k
(b) 1 + k
(c) $$\frac{x^2}{2}$$ + k
(d) log x + k

Question 8.
∫(√x + $$\frac{1}{√x}$$) dx =
(a) $$\frac{1}{x}$$ x$$\frac{1}{3}$$ + 2x$$\frac{1}{2}$$ + c
(b) $$\frac{2}{3}$$ x$$\frac{2}{3}$$ + $$\frac{1}{2}$$x2 + c
(c) $$\frac{2}{3}$$ x$$\frac{3}{2}$$ + 2x$$\frac{1}{2}$$ + c
(d) $$\frac{3}{2}$$ x$$\frac{3}{2}$$ + $$\frac{1}{2}$$x$$\frac{1}{2}$$ + c

Answer: (c) $$\frac{2}{3}$$ x$$\frac{3}{2}$$ + 2x$$\frac{1}{2}$$ + c

Question 9.
∫$$\frac{sin^2x-cos^2x}{sin^2xcos^2x}$$ dx is equal to
(a) tan x + cos x + c
(b) tan x + cosec x + c
(c) tan x + cot x + c
(d) tan x+ sec x + c

Answer: (c) tan x + cot x + c

Question 10.
∫$$\frac{cos 2x dx}{(sinx+cosx)^2}$$ =
(a) –$$\frac{1}{sinx+cosx}$$ + c
(b) log | sin x + cos x | + c
(c) log | sin x – cos x | + c
(d) $$\frac{1}{(sinx+cosx)^2}$$

Answer: (b) log | sin x + cos x | + c

Question 11.
∫$$\frac{dx}{√x}$$ =
(a) √x + k
(b) 2√x + k
(c) x + k
(d) $$\frac{2}{3}$$x3/2 + k

Question 12.
$$\frac{d}{dx}$$∫f(x)dx is equal to
(a) f'(x)
(b) f(x)
(c) f'(x’)
(d) f(x) + c

Question 13.
∫$$\frac{xdx}{(x-1)(x-2)}$$ equals

(d) log |(x – 1)(x – 2) + c

Answer: (b) log|$$\frac{(x-2)^2}{x-2}$$| + c

Question 14.
What is the value of $$\int_{0}^{\pi / 2}$$ $$\frac{\sqrt{tan x}}{\sqrt{tan x}+\sqrt{cot x}}$$ dx?
(a) $$\frac{π}{2}$$
(b) $$\frac{π}{4}$$
(c) $$\frac{π}{8}$$
(d) None of these

Answer: (b) $$\frac{π}{4}$$

Question 15.
What is the value of $$\int_{0}^{\pi / 2}$$ $$\frac{sinx – cos x}{1+sin xcos x}$$ dx?
(a) 1
(b) $$\frac{π}{2}$$
(c) 0
(d) –$$\frac{π}{2}$$

Question 16.
What is the value of $$\int_{\pi / 6}^{\pi / 3}$$ $$\frac{dx}{sin2x}$$?
(a) $$\frac{1}{2}$$ log(-l)
(b) log(- 1)
(c) log 3
(d) log √3

Question 17.
What is the value of $$\int_{-1}^{1}$$ sin³ x cos² xdx?
(a) 0
(b) 1
(c) $$\frac{1}{2}$$
(d) 2

Question 18.
What is the value of $$\int_{1}^{e} \frac{1+\log x}{x}$$ dx?
(a) $$\frac{3}{2}$$
(b) $$\frac{1}{2}$$
(c) e
(d) $$\frac{1}{e}$$

Answer: (a) $$\frac{3}{2}$$

Question 19.
$$\int_{-\pi / 2}^{\pi / 2}$$ sin9 xdx =
(a) -1
(b) 0
(c) 1
(d) None of these

Question 20.
$$\int_{0}^{\pi^{2} / 4} \frac{\sin \sqrt{y}}{\sqrt{y}}$$
(a) 1
(b) 2
(c) $$\frac{π}{4}$$
(d) $$\frac{π^2}{8}$$

Question 21.
∫$$\frac{dx}{1+cos x}$$ =
(a) tan $$\frac{x}{2}$$ + k
(b) $$\frac{1}{2}$$ tan $$\frac{x}{2}$$ + k
(c) 2 tan $$\frac{x}{2}$$ + k
(d) tan² $$\frac{x}{2}$$ + k

Answer: (a) tan $$\frac{x}{2}$$ + k

Question 22.
$$\int_{a}^{b}$$ x5 dx =
(a) tan $$\frac{x}{2}$$ + k
(b) $$\frac{1}{2}$$ tan $$\frac{x}{2}$$ + k
(c) 2 tan $$\frac{x}{2}$$ + k
(d) tan² $$\frac{x}{2}$$ + k

Answer: (a) tan $$\frac{x}{2}$$ + k

Question 23.
$$\int_{0}^{\infty} \frac{1}{1+e^{x}}$$ dx =
(a) log 2
(b) -log 2
(c) log 2 – 1
(d) log 4 – 1

Question 24.
$$\int_{0}^{1}$$ x(1 – x)99 is equal to
(a) $$\frac{1}{10010}$$
(b) $$\frac{1}{10100}$$
(c) $$\frac{1}{1010}$$
(d) $$\frac{11}{10100}$$

Answer: (b) $$\frac{1}{10100}$$

Question 25.
What is the value of $$\int_{0}^{1}$$ $$\frac{d}{dx}$${sin-1($$\frac{2x}{1+x^2}$$)}dx?
(a) 0
(b) π
(c) -π
(d) $$\frac{π}{2}$$

Answer: (d) $$\frac{π}{2}$$

Question 26.
$$\int_{0}^{1}$$ $$\frac{x}{1+x}$$ dx =
(a) 1 – log 2
(b) 2
(c) 1 + log 2
(d) log 2

Answer: (a) 1 – log 2

Question 27.
∫$$\frac{cos 2x- cos 2θ}{cos x – cos θ}$$dx is equal to
(a) 2 (sin x + x cos θ) + C
(b) 2 (sin x – x cos θ) + C
(c) 2 (sin x + 2x cos θ) + C
(d) 2 (sin x – 2x cos θ) + C

Answer: (a) 2 (sin x + x cos θ) + C

Question 28.
∫$$\frac{sin x + cos x}{\sqrt{1+2sin x}}$$ dx =
(a) log(sin x – cos x)
(b) x
(c) log x
(d) log sin (cos x)

Question 29.
∫tan-1 √xdx is equal to
(a) (x + 1)tan-1 √x – √x + C
(b) x tan-1 √x – √x + C
(c) √x – x tan-1 √x + C
(d) √x – (x + 1)tan-1 √x + C

Answer: (a) (x + 1)tan-1 √x – √x + C

Question 30.
∫log10 xdx =
(a) loge 10.x loge ($$\frac{x}{e}$$) + c
(b) log10 e.x loge ($$\frac{x}{e}$$) + c
(c) (x – 1) loge x + c
(d) $$\frac{1}{x}$$ + c

Answer: (b) log10 e.x loge ($$\frac{x}{e}$$) + c

Question 31.
∫($$\frac{cos 2θ – 1}{cos 2θ + 1}$$) dθ =
(a) tan θ – θ + c
(b) θ + tan θ + c
(c) θ – tan θ + c
(d) -θ – cot θ + c

Answer: (c) θ – tan θ + c

Question 32.
∫$$\frac{2dx}{\sqrt{1-4x^2}}$$ =
(a) tan-1 (2x) + c
(b) cot-1 (2x) + c
(c) cos-1 (2x) + c
(d) sin-1 (2x) + c

Answer: (d) sin-1 (2x) + c

Question 33.
Value of ∫$$\frac{dx}{\sqrt{2x – x^2}}$$
(a) sin-1 (x – 1) + c
(b) sin-1 (1 + x) + c
(c) sin-1 (1 + x²) + c
(d) –$$\sqrt{2x-x^2}$$ + c

Answer: (a) sin-1 (x – 1) + c

Question 34.
∫x² sin x³ dx =
(a) $$\frac{1}{3}$$ cos x³ + c
(b) –$$\frac{1}{3}$$ cos x + c
(c) $$\frac{-1}{3}$$ cos x³ + c
(d) $$\frac{1}{2}$$ sin² x³ + c

Answer: (c) $$\frac{-1}{3}$$ cos x³ + c

Question 35.
∫$$\frac{dx}{sin(x-a)sin(x-b)}$$ is equal to
(a) sin(b – a) log |$$\frac{sin (x-b)}{sin(x-a)}$$| + C
(b) cosec (b – a) log |$$\frac{sin (x-b)}{sin(x-b)}$$| + C
(c) cosec (b – a) log |$$\frac{sin (x-b)}{sin(x-a)}$$| + C
(d) sin (b – a) log |$$\frac{sin (x-a)}{sin(x-b)}$$| + C

Answer: (c) cosec (b – a) log |$$\frac{sin (x-b)}{sin(x-a)}$$| + C

Share: