Definite integration practice · Medium

Option A ($\frac{8}{\pi} f(2)$) is correct because it is the result of applying L'Hopital's rule to the limit and evaluating at $x = \frac{\pi}{4}$.

Option A ($\pi \log 2$) is correct because the integral can be simplified using trigonometric identities and properties of logarithms, leading to the result $\pi \log 2$.

Option C ($\frac{1}{3}$) is correct because the determinant simplifies to $\sin 3x$, and the integral of $\sin 3x$ from 0 to $\pi/2$ is $\frac{1}{3}$.

Option D (-5) is correct because the integral of an odd function over a symmetric interval is zero, and using the given integral from 0 to 1, we can find the integral from -1 to 0.

Option C (3π/256) is correct because the integral of sin⁴(x)cos⁶(x) from -π/2 to π/2 evaluates to 3π/256 using the Gamma function.

Option C (-9e) is correct because it is the result of applying integration by parts and simplifying the expression.

Option A ($\pi$) is correct because after differentiating the given integral equation and solving for f(x), we find f(-$\frac{\pi}{3}$) = $\pi$.

Option A ($\frac{{\pi^2}}{{16}}$) is correct because the integral is solved using properties of definite integrals and trigonometric substitutions.

Option B (1) is correct because substituting k = 1 into the area equation derived from the integral satisfies the given condition.

Option C ($\frac{{ - \\,n}}{{m + 1}}\\,,{I_{m,n - 1}}$) is correct because it is the result of applying integration by parts to the given integral.

Option B (1/3) is correct because the area bounded by the parabola and the line is calculated by integrating the difference between the two functions.

Option B ($\frac{\pi }{{5}}$) is correct. The integral is solved by using the property $\int \limits_0^a f(x) dx = \int \limits_0^a f(a-x) dx$ and then integrating by substitution.

Option A ($\frac{9 !}{(3 !)^3}$) is correct because after evaluating the integral and comparing with the given expression, the number of ways to distribute the objects is $\frac{9 !}{(3 !)^3}$.

Option A (4) is correct because after applying the properties of definite integrals and simplifying, the value of k is found to be 4.

Option A ($\frac{\pi^2}{4}$) is correct because the integral is solved using the property of definite integrals and trigonometric substitution.

Option B (0) is correct because the integral simplifies to zero after applying properties of inverse trigonometric functions and definite integrals.

Option C ($x+\log |\sin (x-\frac{\pi}{4})|+c$) is correct because after applying a substitution and simplifying, the integral evaluates to this expression.

The correct answer is D (None of these) because the integral evaluates to 2, which is not among the provided options.

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

Option B ($\frac{{\alpha+1}}{{\beta+1}}$) is incorrect. The correct answer is A ($\frac{{\beta+1}}{{\alpha+1}}$) because it is found by evaluating the limit of the ratio of sums using integrals.

Option A is correct because after integrating by parts and simplifying, the resulting function f(x) and constant A match the values given in option A.

Option B ($\frac{\pi}{2}$) is correct because the integral simplifies using substitution and properties of even and odd functions.

Option C ($\frac{2}{\mathrm{\pi }}\I_1$) is correct because it is derived by using the property of definite integrals and solving for I2 in terms of I1.

Option D (${a_1}$ and ${a_3}$) is correct because only the integrals involving $\cos^2 x \sin x$ and $\sin^3 x$ are non-zero over the interval $[0, \pi]$.

The correct answer is C ($\frac{1}{2}$) because after applying Leibnitz rule and integration, F(e) is found to be $\frac{1}{2}$.

Option C (${I} < 1$) is correct because the integrand $\frac{{1}}{{1+x^2}}$ is less than 1 over the interval [0, 1], thus the integral must be less than 1.

Option D ((x-4)²+(y-5)²=25) is incorrect. Option D is incorrect because the incentre is (1,1) which does not lie on the circle (x-4)²+(y-5)²=25. The incentre of the triangle formed by the lines xy=0 and x/3+y/4=1 is (1,1).

Option A ($g(x)$ must be differentiable at $x=a$) is correct because for $f(g(x))$ to be differentiable at $x=a$, $g(x)$ must also be differentiable at $x=a$.

Option C (1) is correct because by comparing the integral with simpler integrals, we can bound its value and determine that the greatest integer function of twice the integral is 1.

Option C ($\frac{{n!}}{{{a^{n + 1}}}}$) is correct because it is obtained by differentiating the given integral n times with respect to 'a'.