Definite integration practice · Easy

The correct answer is B (2) because it satisfies the given equation after evaluating the integral and applying the inverse secant function.

Option B ($a+1-\frac{e}{2}$) is correct because it uses integration by parts and the given integral value to find the value of the second integral.

Option B ($\frac{2}{3}$) is correct because it is the result of applying L'Hopital's rule and Leibniz's rule to evaluate the limit.

Option C ($a$) is correct because the integral simplifies to 'a' after applying the property of definite integrals.

Option B ($4 \sqrt{3}-4-\frac{\pi}{3}$) is correct because it is obtained by simplifying the integral using trigonometric identities and evaluating the resulting integral.

Option B, $\frac{\pi^2}{2}$, is correct because it results from evaluating the integral using the given function and its derivative.

Option C (2+log(2/(e²+1))) is correct because it is the result of integrating the function over the given interval and dividing by the length of the interval.

Option D (0) is correct because the integral of sin¹⁰⁰(x) from 0 to pi/2 is equal to the integral of cos¹⁰⁰(x) from 0 to pi/2, resulting in a difference of 0.

Option B (1/4 log 3) is correct because after substitution and integration, the definite integral evaluates to (1/4)log 3.

Option B ($\frac{{1}}{{2}}\,\left( {b - a} \right)$) is correct because it is derived by using the property of definite integrals where $\int \limits_a^b f(x) dx = \int \limits_a^b f(a+b-x) dx$.

Option A ($\pi$) is correct because the integral evaluates to $\pi$ for all n in N.

Option B ($I_1 > I_2$) is correct because $2^{x^3} < 2^{x^2}$ for $0 < x < 1$, implying the integral of $2^{x^2}$ from 0 to 1 is greater.

Option A (Continuous and non differentiable at $x = 2$) is correct because the derivative is an absolute value function, which is continuous but not differentiable at the point where the expression inside the absolute value is zero.

Option C ($\frac{{\pi}}{{2}}$) is correct because using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ simplifies the integral to $\frac{{\pi}}{{2}}$.

Option C is correct because the integral is solved using the property of definite integrals and adding the original and transformed integrals.

Option C (-5) is correct because it uses the properties of definite integrals to manipulate and solve for the desired integral value.

Option B (${\pi ^2}$) is correct because the integral simplifies to a solvable form using properties of definite integrals and trigonometric substitutions.

Option A ($\mathrm{0}$) is correct because the integrand is an odd function, and the integral is taken over a symmetric interval.

Option C ($2 \sqrt{{2}}-1$) is correct because it represents the area calculated by integrating the difference between the sine and cosine functions over the given interval, considering their intersection point.

Option C ($\mathrm{lo}{\mathrm{g}_e}2 + \frac{\pi }{2} - 1$) is correct because it is the result of evaluating the definite integral using integration by parts and trigonometric substitution.

Option B (| b\ | - | a\ |$\left| {\,b\} \right|\, - \,\left| {\,a\} \right|$) is correct because the integral of |x|/x from a to b, where a < b, always results in the absolute value of b minus the absolute value of a, regardless of the signs of a and b.

Option B is correct because the function $x-[x]$ is periodic with period 1, so the integral simplifies to $n \int_0^1 x dx = n/2$.

Option D ($\frac{\pi}{4}$) is correct because the integral can be solved using the property of definite integrals and trigonometric identities.

Option B ($\frac{\mathrm{\pi }}{4}$) is correct because after substituting $x = \cos 2\theta$, the integral simplifies and evaluates to $\frac{\mathrm{\pi }}{4}$.

Option B ($- 1 + \sqrt 5$) is correct because the limit of the Riemann sum can be expressed as a definite integral, which evaluates to $-1 + \sqrt{5}$.

Option B (0) is correct because the greatest integer function of x in the interval [0, 0.9] is 0, making the integral equal to 0.

Option A ($\log 2$) is correct because the limit of the given sum can be expressed as a definite integral, which evaluates to log 2.

Option A ($\frac{1}{4} \log \frac{3}{2}$) is correct because the integral can be solved using the property of definite integrals and substitution.

Option B ($\frac{[x]}{\log 2}$) is correct because the integral simplifies to this expression using the properties of the greatest integer function.

Option B (1) is correct. By evaluating the integral $I_n + I_{n+2}$ and then taking the limit as $n$ approaches infinity, the result is 1.