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 →Integrate the function: $\frac{\cos x}{\sqrt{4-\sin ^{2} x}}$
View full solution →Integrate the function $\int {\frac{{{e^{5\log x}} - {e^{4\log x}}}}{{{e^{3\log x}} - {e^{2\log x}}}}} dx$
View full solution →Integrate the function $\frac{\sin x}{\sin (x-a)}$
View full solution →Integrate the function $\frac{5 x}{(x+1)\left(x^{2}+9\right)}$
View full solution →Integrate the function $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$ [Hint: $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=\frac{1}{x^{\frac{1}{3}}\left(1+x^{\frac{1}{6}}\right)}$ Put $x = t^6$]
View full solution →Evaluate the integral $\int_{-1}^{1} \frac{d x}{x^{2}+2 x+5}$ using substitution.
View full solution →Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^{2} x} d x$ using substitution.
View full solution →Evaluate the integral $\int_{0}^{1} \frac{x}{x^{2}+1} d x$ using substitution.
View full solution →Evaluate the definite integral $\int_0^1 \frac{d x}{\sqrt{1-x^2}}$
View full solution →Evaluate the definite integral $\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{4}} {\cos ecxdx} $
View full solution →Evaluate the integral $\int\limits_1^2 {\left( {\frac{1}{x} - \frac{1}{{2{x^2}}}} \right){e^{2x}}dx} $ using substitution.
View full solution →Evaluate the integral $\int_{0}^{2} x \sqrt{x+2}$ (Put $x + 2 =t^2$) using substitution.
View full solution →Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^{5} \phi~ d \phi$ using substitution.
View full solution →Evaluate the definite integral $\int\limits_2^3 {\frac{{xdx}}{{{x^2} + 1}}} $
View full solution →Integrate the function $\sqrt {1 + \frac{{{x^2}}}{9}} $
View full solution →Evaluate the integral $\int_{0}^{2} \frac{d x}{x+4-x^{2}}$ using substitution.
View full solution →Evaluate the integral $\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$ using substitution.
View full solution →Evaluate the definite integral $\int _ { 0 } ^ { 1 } x e ^ { x ^ { 2 } } d x.$
View full solution →Integrate the function $\sqrt{1+3 x-x^{2}}$
View full solution →Integrate the rational function $\frac{3 x+5}{x^{3}-x^{2}-x+1}$
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 →