Integrals Online Objective Test | Integrals Online Objective Test

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q1.

\int \frac{1}{\sqrt{1 - x}} d x

is equal to

2 \sqrt{1 - x} + C

2 \sqrt{1 - x} + C

\sqrt{1 - x}

+ C

2 \sqrt{1 - x} + C

2 \sqrt{1 - x} + C

D. 3x−2−−−−√+C

Answer : Option A
Explaination / Solution:

\int (1 - x)^{- 1 / 2} d x

= \frac{{(1 - x)}^{1 / 2}}{(1 / 2) (- 1)}

+ C

= - 2 \sqrt{1 - x}

+ C

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q2. An antiderivative of

\frac{x}{\cos^{2} x}

is equal to

A. Cot x + C

B. x tanx + C

C. x tanx + log |cosx| + C

D. x cosx + log |tanx| + C

Answer : Option C
Explaination / Solution:

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q3. ∫log x dx is equal to

\frac{1}{2} (\log x)^{2}

+ C

B. x + logx + C

C. x log x – x + C

D. 1/x + C

Answer : Option C
Explaination / Solution:

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q4. If f (x) be a function such that

\frac{d}{d x} (f (x)) = \log x,

then f (x) is equal to

x \log (\frac{e}{x})

+ C

x \log (\frac{e}{x})

+ C

\frac{\log x}{x}

+ C

D. x log (x e) + C

Answer : Option A
Explaination / Solution:

Integrating both sides, we get

\int \frac{d}{d x} (f (x)) d x = \int \log x d x w e g e t f (x) = \int \log x d x

$+ C$

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q5.

\int \frac{f^{'} (x)}{\sqrt{f (x)}} d x

is equal to

A. f(x) + C

B. log ( f(x) ) + C

\frac{1}{2} \log | f (x) |

+ C

D. 2√f(x) + C

Answer : Option D
Explaination / Solution:

$\int \frac{f^{'} (x)}{f (x)} d x = \int {(f (x))}^{- 1 / 2} f^{'} (x) d x = \frac{{(f (x))}^{1 / 2}}{1 / 2} + C = 2 \sqrt{f (x)} + C$

$s u b s t i t u e f (x) = t^{2} t h e n f^{'} (x) d x = 2 t d t \Rightarrow \int 2 d t = 2 t + C = 2 \sqrt{f (x)} + C$

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q6.

\int \log (\frac{1}{x} - 1)

dx is equal to

A. 12log(2−x)+C

B. log 2x + C

C. (x−1)log(1−x)+1−xlnx+C

D. (3x+1)log(2+x)+1−xln2x+C

Answer : Option C
Explaination / Solution:

∫log(1−x)dx−∫logx dx⇒(x−1)log(1−x)+1−xlnx+C

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q7.

\int (\log (\log x) + {(\log x)}^{- 1})

dx is equal to

A. x log (log x)

- \frac{x}{\log x}

+ C

B. x−log2x−1x+2+C

C. x log (log x) +

\frac{x}{\log x}

+ C

D. x log (log x) + C

Answer : Option D
Explaination / Solution:

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q8. Which of the following is not equal to

\int \frac{- 1}{\sqrt{1 - x^{2}}} d x

\cos^{- 1} x

+ C

B. -

\cos^{- 1} x

+ C

C. −sin−1x+C

D. (secx)(cos−1x)(cosx)+C

Answer : Option B
Explaination / Solution:

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q9.

$\int (1 - \cos x)$ $\cos e c^{2} x$ dx is equal to

A. 3cot2x3+C

B. cotx2+C

C. tanx2+C

D. 12tanx+C

Answer : Option C
Explaination / Solution:

# Integrals # CBSE 12th Mathematics Prepare / Learn

Q10.

\int \sqrt{x^{2} + a^{2}} d x

is equal to

A. $Put the Modulus sign in the input of logarithmic function$

Standard Formulae Can be done by By-Part taking '1' as the second function and

\int \sqrt{x^{2} + a^{2}} d x

as first function.

B. tan−1(x2+x+2)+C

Answer : Option A
Explaination / Solution:

$Put the Modulus sign in the input of logarithmic function$

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