Definite integration practice · Mixed difficulty

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 ($2\,\, + \,\,{\log _e}\left( {\frac{{15}}{7}} \right)$) is correct because evaluating the definite integral using partial fractions yields this result.

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 B (-2 log 3) is correct because the integral simplifies using trigonometric identities and evaluates to -2 log 3 after applying the limits.

Option C ($\pi-2$) is correct because the integral is evaluated by splitting the integral and using the periodicity of the sine function.

Option D ($-\frac{3}{4}$) is correct because it is the result of integration by parts, using the given information about the function's intercepts and area.

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 C ($\left( {1/7} \right)\log \left( {6/5} \right)$) is correct because the integral is solved by using a substitution to simplify the expression.

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 B ($e^6-e^3$) is correct because the integral is solved by breaking it into intervals based on the greatest integer function.

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 D ($\mathrm{0}$) is correct because the integrand is an odd function integrated over a symmetric interval.

Option C ($\frac{\mathrm{32}}{\mathrm{9}}$) is correct because after differentiating and substituting, the value of $F'(4)$ is found to be $\frac{\mathrm{32}}{\mathrm{9}}$.

Option B ($\mathcal{F}_{n+1}-\mathcal{F}_n=I_{n+1}$) is correct because it follows from the derivation of the difference between consecutive terms of $\mathcal{F}$ and its relation to $I$.

Option B ($\frac{\mathrm{\pi }}{2} - 1$) is correct because the integral is solved by simplifying the expression and using trigonometric substitution.

Option B ($\frac{4}{3}$) is correct because the integral simplifies to an integral of sin(x)sqrt(cos(x)) which can be solved by substitution.

Option B ($\frac{5}{2}$) is correct because the integral is evaluated by splitting it into two parts based on the sign of the logarithm and then integrating.

Option D (28/3) is correct because the integral of the absolute value function |1-x²| is computed by splitting the integral into intervals where the function inside the absolute value is positive or negative, and then integrating each part separately.

Option D ($(1 / 8) \log 3$) is correct because it is the result of evaluating the definite integral after a suitable substitution.