INTEGRALS - Rajasthan Board (RBSE) STD 12 Science MATHS Questions

$\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sec^2\text{x dx}$ is equal to:

-1
0
1
2

Choose the correct answer in Exercise:
The value of the integral $\int^{1}_{\frac{1}{3}}\frac{(\text{x}-\text{x}^{3})^{\frac{1}{3}}}{\text{x}^{4}}\text{dx}\ \text{is}$

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Q 3M.C.Q (1 Marks)1 Mark

$\int\frac{(1+\text{log x})^2}{1+\text{x}^2}\text{dx}=$

$\frac{1}{3}(1+\text{log})^3+\text{c}$
$\frac{1}{2}(1+\text{log})^2+\text{c}$
$\log(\text{log }1+\text{x})+2$
$\text{None of these}$

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Q 4M.C.Q (1 Marks)1 Mark

The value of the integral $\int\limits^2_{-2}\big|1-\text{x}^2\big|\text{dx}$ is:

4
2
-2
0

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Q 5M.C.Q (1 Marks)1 Mark

$\int\frac{\text{x}^4+1}{\text{x}^2+1}\text{ dx}$ is equal to:

$\frac{\text{x}^3}{3}+\text{x}+\tan^{-1}\text{x}+\text{c}$
$\frac{\text{x}^3}{3}-\text{x}+\tan\text{x}+\text{c}$
$\frac{\text{x}^3}{3}+\text{x}+2\tan^{-1}\text{x}+\text{c}$
$\frac{\text{x}^3}{3}-\text{x}+2\tan^{-1}\text{x}+\text{c}$

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Q 6Assertion (A) & Reason (B) MCQ1 Mark

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.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

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Q 7Assertion (A) & Reason (B) MCQ1 Mark

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.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

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Q 8Assertion (A) & Reason (B) MCQ1 Mark

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.
$\frac{\text{n-1}}{\text{n}}\frac{\text{n}-3}{\text{n}-2}\frac{\text{n-5}}{\text{n-4}}...\frac{1}{2}\frac{\pi}{2}$

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.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

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Q 9Assertion (A) & Reason (B) MCQ1 Mark

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.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

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Q 10Assertion (A) & Reason (B) MCQ1 Mark

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 $\text{x}$

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.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

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Q 111 Marks Question1 Mark

If $\int\limits^{\text{a}}_0\frac{1}{4+\text{x}^2}\text{dx}=\frac{\pi}{8}$, find the value of a.

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Q 121 Marks Question1 Mark

Find:
$\int \frac{\sin^{2} \text{x} - \cos^{2} \text{x}}{\sin \text{x} \cos \text{x}} \text{dx}$

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Q 131 Marks Question1 Mark

Evaluate: $\int\limits_{\text{e}}^{\text{e}^2}\frac{\text{dx}}{\text{x}\log\text{x}}$

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Q 141 Marks Question1 Mark

If $\text{f}(\text{x}) = \int\limits_{0}^{\text{x}}\text{t}\sin\text{t }\text{dt},$then write the value of f'(x).