Definite integration practice · Medium

Option B ($\frac{{3}}{{4}}$(2)^{{4/3}}-$\frac{{3}}{{4}}$) is correct because the limit of the sum can be expressed as a definite integral, which can then be evaluated.

Option C ($\pi / 12$) is correct because the integral can be solved by using the property $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ and adding the two integrals.

Option D ($\pi \left( {\beta - \alpha } \right)$) is incorrect. Option B ($\frac{\pi }{2}\,\left( {\beta - \alpha } \right)$) is correct because the integral evaluates to $\frac{\pi}{2}(\beta - \alpha)$ after substitution and simplification.

Option A ($\frac{\pi}{{12}}$) is correct because using the property of definite integrals and adding the two forms of the integral leads to a simple result.

Option B ($I_{2013}+(2013) I_{2012}$) is correct because it is derived from the recurrence relation obtained by integration by parts.

The correct answer is 7 because solving the integral equation gives the quadratic function, and finding the range of 'a' for real roots leads to 7 integral values.

Option A (4) is correct because the product of the perpendicular distances from the foci to any tangent of the ellipse equals the square of the semi-minor axis, and the integral of the fractional part function is then evaluated.

Option C (100) is correct because the function has a period of 4, and the integral from 10 to 50 is 10 times the integral over one period.

Option B ($\pi^2$) is correct. By splitting the integral and using the property of even and odd functions, and then applying the substitution method, the integral evaluates to $\pi^2$.

Option C ($2x\,\sqrt {1\,\, + \,\,{{{x^4}}}}$) is correct because it is the result of applying the Leibniz rule for differentiating under the integral sign.

Option B ($1-\frac{{1}}{{\sqrt{{2}}}}$) is correct because the integral simplifies to $\int_0^{\pi/4} \sin x dx$ due to the greatest integer function being zero in the interval.

Option A is correct because the integral can be rewritten and evaluated using the given definitions of α, β, and P_n(x).

Option C ($\operatorname{Tan}^{-1} 1 / 2$) is correct because after applying the appropriate substitutions and evaluating, the integral results in $\operatorname{Tan}^{-1} 1 / 2$.

Option C (2 : 1) is correct because the ratio ${I_n}\,:\,{I_{n + 1}}$ simplifies to 2:1 after applying trigonometric identities and integration by parts.

Option D is correct because using trigonometric identities and substitution, the integral simplifies to 1/6.

Option C ($- \,\frac{a}{\pi }\\,\ln \\,2$) is correct because after evaluating the definite integral and using properties of even and odd functions, we find that $c = - \\,\frac{a}{\pi }\\,\ln \\,2$.

Option B (2) is correct because the absolute value simplifies to 1 within the given interval, and the integral of 1 over the interval [-1, 1] is 2.

Option C (2) is correct because the integral of |1-x| from -1 to 1 is calculated by recognizing that 1-x is positive over the entire interval.

Option D ($\frac{1}{2}$) is correct because the limit is evaluated using L'Hopital's rule and the Leibniz rule for differentiation under the integral sign.

Option D ($4\sqrt 2$) is correct because after evaluating the integral of the absolute value of the difference between cosine and sine functions over the interval $[0, 2\pi]$, the result is $4\sqrt{2}$.

Option D ($\pi / 6$) is correct because the limit of the given sum can be expressed as a definite integral, which evaluates to $\pi/6$.

Option B (I2 > I1) is correct because for x in [1, 2], 1/√(1+x²) < 1/x, implying that the integral I2 is greater than I1.

Option B ($- \pi \log \\,2$) is correct because the integral is solved using trigonometric identities and properties of logarithms and definite integrals.

Option A ($\frac{\pi}{4}$) is correct because the derivative of the integral is zero, implying the integral is a constant, which can be evaluated at a specific point.

Option A (1) is correct because evaluating the definite integral of g(x)-f(x) from 0 to 3 gives a result of 1.

Option B ($\tan ^{-1} 2+\frac{1}{3} \tan ^{-1} 8-\frac{\pi}{3}$) is correct because it is the result of evaluating the definite integral after simplification and applying trigonometric identities.

Option C is correct because the integral can be split into two parts and evaluated, resulting in $\frac{1}{2} \log_{10} e + 1$.

Option B (1/8) is correct because after evaluating the definite integral of cos⁴(x) from 0 to pi/3, the resulting expression matches the form a*pi + b*sqrt(3), and a² + b equals 1/8.

Option A ($\frac{1}{9} < {I^2} < \frac{1}{8}$) is correct because by finding the bounds of the integrand, we can determine the bounds of the integral and thus its square.

Option C ($\frac{{32 \sqrt{2}}}{\pi^3}$) is correct because after integration by parts, the limit as x approaches 0 of $x^3 \sin(1/x)$ is 0, leading to the correct result.