Definite integration practice · Mixed difficulty

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

Option B (1 - 1/√2) is correct because the integral simplifies to the integral of sin(x) from 0 to π/4, which evaluates to 1 - 1/√2.

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

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

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 C ($\log _{e} 2$) is correct because the limit can be expressed as a definite integral which evaluates to $\log_e 2$.

Option C ($\log _e 2$) is correct because the limit of the sum can be expressed as a definite integral.

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 ($\frac{44-\sqrt{3}}{\sqrt{2}}$) is correct because the integral is evaluated by breaking it into intervals where cos(x) is either positive or negative.

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

Option A (0) is correct because the integrand is an odd function and the limits of integration are symmetric about the origin.

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 $[0, \pi]$.

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 B (4/3) is correct because the integral evaluates to log(4/3), thus K = 4/3.

Option A (${e^{2x}}\sin \\,3x\\, + \\,c$) is correct because the integral can be solved by recognizing a pattern where the cosine term cancels out after integration by parts.

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

Option D ($\frac{1}{11}-\frac{3}{12}+\frac{3}{13}-\frac{1}{14}$) is correct because using the property of definite integrals and expanding the polynomial simplifies the integral to the given expression.

Option A (1) is correct because the integral of the fractional part function from -1 to 1 equals 1.

Option C ($\left( {\log \,2,\,\infty } \right)$) is correct because the inequality leads to ${e^\alpha } > 2$, which implies $\alpha > \log 2$.

Option C is correct because the integral is split into two parts, and the first part is zero due to the integrand being an odd function, leaving only the second part to be evaluated.

Option A (e) is correct because the function f(x) is found to be $e^{x^2}$, and thus f(1) = e.

Option D is correct. The integral is solved by using substitution and simplifying the expression, resulting in (1/8)log 3.

Option B ($\frac{{\pi}}{{2}}-1$) is correct because by rationalizing the denominator and using basic integration formulas, the integral evaluates to this value.

Option B ($2-\sqrt{2}$) is correct because the integral of the greatest integer function of x squared from 0 to 1.5 is equal to $2-\sqrt{2}$.