Sample QuestionsINTEGRALS questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
$\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sec^2\text{x dx}$ is equal to:
Answer: D.
View full solution →The value of $\int\frac{\cos2\text{x}}{{\cos}{\text{ x}}}\text{dx}$ is equal to:
- ✓
$2\sin\text{x}-\ell\text{ n }\mid\sec\text{x}+\tan\text{x}\mid+\text{ c}$
- B
$2\sin\text{x}-\ell\text{ n }\mid\sec\text{x}-\tan\text{x}\mid+\text{ c}$
- C
$2\sin\text{x}+\ell\text{ n }\mid\sec\text{x}+\tan\text{x}\mid+\text{ c}$
- D
$3\sin\text{x}-\ell\text{ n }\mid\sec\text{x}+\tan\text{x}\mid+\text{ c}$
Answer: A.
View full solution →Choose the correct option from given four options:
$\int\frac{\text{x}}{\text{x}+1}$ is equal to:
- A
$\text{x}+\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
- B
$\text{x}+\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
- C
$\text{x}-\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
- ✓
$\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
Answer: D.
View full solution →$\int\limits^1_0\frac{\text{d}}{\text{dx}}\Big\{\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)\Big\}\text{dx}$ is equal to:
- A
$0$
- B
${\pi}$
- ✓
$\frac{\pi}{2}$
- D
$\frac{\pi}{4}$
Answer: C.
View full solution →$\int\limits^\sqrt{3}_1\frac{1}{1+\text{x}^2}\text{ dx}$ is equal to:
- ✓
$\frac{\pi}{12}$
- B
$\frac{\pi}{6}$
- C
$\frac{\pi}{4}$
- D
$\frac{\pi}{3}$
Answer: A.
View full solution →Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: $\int_{0}^{2\pi}\sin^3\text{x}\text{ dx}=0$
Reason: $\sin^3\text{x}$ an odd function.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- ✓
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: B.
View full solution →Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The value of $\int_{0}^{\frac{\pi}{0}}\sin^6\text{xdx}=\frac{5\pi}{16}$
Reason: If $\text{n}$ is even, then $\int_{0}^{\frac{\pi}{0}}\sin^\text{n}\text{xdx}$ equals.
$\frac{\text{n-1}}{\text{n}}\frac{\text{n}-3}{\text{n}-2}\frac{\text{n-5}}{\text{n-4}}...\frac{1}{2}\frac{\pi}{2}$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
Answer: D.
View full solution →Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The function $F(x)$ satisfies $\text{F(x}+\pi)=\text{F}\text{(x)}$ for all real $\text{x}$
Reason: $\text{Sin}^2(\text{x}+\pi)=\sin^2\text{x}$ for all real $\text{x}$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
Answer: D.
View full solution →Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices :
Assertion : $\int\sin3\text{x}\cos5\text{x}\text{ dx}=\frac{-\cos8\text{x}}{16}+\frac{\cos2\text{x}}{4}+\text{C}$
Reason : $2\cos\text{A}\sin\text{B}=\sin(\text{A+B})-\sin(\text{A-B})$
- ✓
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: A.
View full solution →Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices :
Assertion : $\text{I}=\int_{0}^{1}\frac{\text{dx}}{3\sqrt{1+\text{x}^3}}=\int_{0}^{{2}^\frac{-1}{3}}\frac{\text{dt}}{1-\text{t}^3}$
Reason : The integrand of the integral I becomes rational by the substitution $\text{t}=\frac{\text{x}}{3\sqrt{1+\text{x}^3}}$
- ✓
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: A.
View full solution →If $\int\limits^{\text{a}}_0\frac{1}{4+\text{x}^2}\text{dx}=\frac{\pi}{8}$, find the value of a.
View full solution →Find:
$\int \frac{\sin^{2} \text{x} - \cos^{2} \text{x}}{\sin \text{x} \cos \text{x}} \text{dx}$
View full solution →Evaluate: $\int\limits_{\text{e}}^{\text{e}^2}\frac{\text{dx}}{\text{x}\log\text{x}}$
View full solution →Evaluate: $\int\limits_{2}^{4}\frac{\text{x}}{\text{x}^{2} +1}\text{dx}.$
View full solution →If $\text{f}(\text{x}) = \int\limits_{0}^{\text{x}}\text{t}\sin\text{t }\text{dt},$then write the value of f'(x).
View full solution →Find:
$\int\frac{\text{dx}}{5 - \text{8x - x}^{2}}$
View full solution →Find:
$\int \frac{\text{dx}}{\sqrt{3 - \text{2x - x}^{2}}}$
View full solution →Find $\int \frac{\text{d}x}{x^{2} + 4x + 8}$
View full solution →Evaluate: $\int\frac{\cos2\text{x}+2\sin^2\text{x}}{\cos^2\text{x}}\text{dx}$
View full solution →Find: $\int\sin\text{x}.\log\cos\text{x}\text{dx}.$
View full solution →Evaluate:
$\int\text{x log 2x dx}$.
View full solution →Evaluate:
$\int\sin\text{4x}\cos\text{3x dx}$.
View full solution →Evaluate:
$\int\frac{\text{2x.tan-1(x}^{2})}{\text{1 + x}^{4}}\text{dx}. $
View full solution →Evaluate:$\int \frac{1 + x^{2}}{1 + x^{4}} \text{dx}$
View full solution →Evaluate: $\int \cos \text{4 x} \cos 3\text{x dx}$
View full solution →Find:
$\int\frac{x^{2}}{x^{4} + x^{2} - 2}dx$
View full solution →Evaluate: $\int\limits^{\frac{\pi}{2}}_{0}\frac{\sin^{2}x}{\sin x + \cos x}dx$
View full solution →Evaluate:
$\int\limits^{\pi/4}_{0}\bigg(\frac{\sin \text{x} +\cos \text{x}}{3 + \sin 2\text{x}}\bigg)\text{dx}$
View full solution →Evaluate:
$\int\limits^{\pi}_{0} \frac{\text{x} \tan \text{x}}{\sec \text{x} + \tan \text{x}}\text{dx}$
View full solution →Evaluate: $\int\limits_{0}^{\pi}\frac{4\text{x}\sin\text{x}}{1 + \cos^{2}\text{x}}\text{dx}.$
View full solution →Fill in the blanks:
$\int\frac{\text{x}+3}{(\text{x}+4)^2}\text{e}^\text{x}\text{dx}=$ ________.
View full solution →Fill in the blanks:
$\int\limits^{\frac{\pi}{2}}_0\cos\text{x e}^{\sin\text{x}}\text{dx}$ is equal to_________.
View full solution →Fill in the blanks:
If $\int\limits^\text{a}_0\frac{1}{1+4\text{x}^2}\text{dx}=\frac{\pi}{8},$ then a = ________.
View full solution →Fill in the blanks:
$\int\frac{\sin\text{x}}{3+4\cos^2\text{x}}\text{dx}=$ ________.
View full solution →Fill in the blanks:
The value of $\int\limits^\pi_{-\pi}\sin^3\text{x}\cos^2\text{x dx}$ is _______.
View full solution →