Definite integration practice · Medium

Option D (I2 < I1) is correct because for x in [0, 1], 2^(x3) < 2^(x2), implying that the integral I2 is less than I1.

Option D ($a=\frac{{4}}{{\pi}}, b=0$) is correct. By using the given derivative and integral conditions, we can solve for the constants a and b.

Option D (1) is correct because it is found by evaluating the integral of the difference between g(x) and f(x) over the given interval.

Option D (8 + $\frac{3 \sqrt{3}}{2} \pi$) is correct because after applying the appropriate substitutions and evaluating, the integral results in 8 + $\frac{3 \sqrt{3}}{2} \pi$.

Option B ($\frac{\pi}{2}$) is correct because the integral of $f(x-\frac{1}{x})$ from 0 to infinity is equal to the integral of $f(x)$ from 0 to infinity when $f(x)$ is an even function.

Option B is correct because the integral can be solved using the property $\int_a^b \frac{f(x)}{f(x)+f(a+b-x)} dx = \frac{b-a}{2}$.

Option B (2) is correct because it is found by using integration by parts twice and simplifying the resulting expression.

Option A is correct because integrating by parts simplifies the integral, leading to the result $2 \tan \frac{3 \pi}{8}$.

Option C ($75\pi$) is correct because it is the result of evaluating the definite integral using properties of definite integrals.

Option B ($\frac{\pi}{2} \int\limits_{0}^\pi f(\sin x) d x$) is correct because it applies the property of definite integrals where $\int_0^a x f(sin(x))dx = \frac{a}{2}\int_0^a f(sin(x))dx$.

Option C ($\frac{32}{3}$ sq. units) is correct because the area is calculated by integrating the difference between the curves from y = 0 to y = 4 and multiplying by 2 due to symmetry.

Option A ($2(\sqrt{{2}}+1)$) is correct because the integral is solved by separating it into two parts and using integration by parts on the second part.

Option C is correct because after converting the product to a summation and evaluating the limit using integration, the result is $\frac{2}{{{e^2}}}\,\,{e^{\pi /2}}$.

Option A, $f(x)$ is an increasing bounded function for all $x \in R$, is correct because solving the differential equation derived from the given equation yields $f(x) = ke^x$, which is an increasing function. However, the solution provided in the original response is incorrect, as $f(x) = ke^x$ is unbounded.

Option C is correct because the integrand is continuous in the given interval, and thus the integral is differentiable.

Option A ($\frac{3}{8}$) is correct because the definite integral evaluates to 3/8 after using a trigonometric substitution.

Option C ($5-\sqrt{2}-\sqrt{3}$) is correct because it is the result of evaluating the definite integral of the greatest integer function of x squared over the interval [0, 2].

Option A ($50 I_6-9 I_5=x I_5^{\prime}$) is correct because it is derived from integration by parts and simplification of the given integral $I_n(x)$.

The correct answer is D ($\frac{4}{3}$) because after evaluating the integral of the absolute value of cosine cubed from 0 to pi, the result is $\frac{4}{3}$.

Option B ($6(\frac{\pi}{2})^5-120(\frac{\pi}{2})^3$) is correct because after applying integration by parts twice, the expression $I_6 - 360I_2$ simplifies to $6(\frac{\pi}{2})^5-120(\frac{\pi}{2})^3$.

Option C (The value of $r$) is correct because the integral depends only on the constant term $r$ due to the odd function property of the logarithmic terms.

Option C is correct because the integral evaluates to the difference in absolute values at the bounds, which is a direct consequence of the integrand being |x|/x.

Option C ($\frac{\pi}{2}-1$) is correct because it is the result of integrating the given definite integral after rationalizing the denominator.

Option A (0) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval.

Option B (Only (2)) is correct because statement 2 is true after evaluating the integral, while statement 1 is false.

Option C ($\frac{{5-\sqrt{{5}}}}{{2}}$) is correct because the integral of the greatest integer function is evaluated by splitting the interval based on where the function takes integer values.

Option A (3) is correct because differentiating both sides using Leibniz's rule and then substituting $x$ such that $\sin x = \frac{1}{\sqrt{3}}$ gives the value of $f(\frac{1}{\sqrt{3}})$.

Option C is correct because the expression simplifies to $\frac{{4 - 3\,\,\sqrt 3 }}{3}$ after evaluating the integrals using the given information about the derivatives.

Option A ($x \tan \frac{x}{2} + c$) is correct because the integral can be solved by rewriting the integrand using trigonometric identities and then integrating by parts.

Option B ($\pi$) is correct because it is the value obtained after evaluating the definite integral using trigonometric substitution.