Definite integration practice · Easy

Option A (-1/2) is correct because the integral simplifies to the integral of [x] from -1/2 to 1/2, which is -1/2.

Option B (4) is correct because by setting up ratios of the combinations and solving the resulting equations, we can find the value of r.

Option B ($\frac{\mathrm{\pi}}{6}$) is correct because the integral can be solved by direct substitution and evaluating the limits.

Option C ($3I$) is correct because the integral $\int_3^{3+3T} f(2x) dx$ simplifies to $3I$ using the periodicity of the function $f(x)$.

Option D ($k$) is correct because the integral of odd powers of x from -3 to 3 is zero, leaving only the integral of the constant k.

The correct answer is B (${\log _e}\,16$) because it is the result of simplifying the integral and evaluating it within the given limits.

Option B (0) is correct because applying L'Hopital's rule to the limit results in an expression that simplifies to 0 as x approaches 0.

Option B ($\frac{\pi}{2}$) is correct because it satisfies the integral equation after applying properties of definite integrals.

Option D (28/3) is correct because it involves splitting the integral into intervals where $|1-x^2|$ is either $(1-x^2)$ or $(x^2-1)$, then integrating and summing the results.

Option B is correct because the integral is evaluated by splitting it into two parts based on the sign of the logarithm function and then using a simple substitution.

Option C ($\frac{1}{6}$) is correct because the integral can be simplified by substitution, leading to an arctangent integral which evaluates to the given value.

Option C ($\mathrm{19}$) is correct. The integral is evaluated by summing the integrals over the intervals [n, n+1] from -2 to 3.

Option A ($1+\frac{4}{\sqrt{3}}$) is correct because it uses the Fundamental Theorem of Calculus and trigonometric identities to find the value of the function at the specified point.

Option D (4) is correct because the integral of the floor function |[x]| from -2 to 2 is equal to 4.

Option D ($\frac{1}{7}$) is correct because the sum of the integrals simplifies to a single integral that can be easily evaluated.

Option D (none of these) is correct because the integral evaluates to 0 due to the integrand being an odd function over a symmetric interval.

Option B ($\frac{{2\pi}}{3}$) is correct because the integral can be solved using the property of definite integrals and trigonometric identities to simplify the expression.

Option B is correct. The integral is solved using substitution and simplifying the expression, resulting in 4 - $\pi$.

Option B ($\frac{\pi}{2}$) is correct because using the property $\int_{-a}^a f(x) dx = \int_{-a}^a f(-x) dx$ simplifies the integral.

The correct answer is C ($\frac{1}{{n + 1}}\\ - \frac{1}{{n + 2}}$) because it is the result of evaluating the definite integral using substitution or Beta function properties.

Option C ($\mathrm{1}\,\mathrm{/}\,\mathrm{4}$) is correct because the integral evaluates to $\frac{1}{4}\,\log \left( {\frac{{3 + 2\,\sqrt 3 }}{3}} \right)$, thus $k = \frac{1}{4}$.

Option C (0) is correct because the integral evaluates to zero due to the given functional equation and properties of definite integrals.

Option A (7) is correct because by converting the summation to an integral and evaluating the limit, we find that $a = 7$ satisfies the given equation.

Option D ($\frac{1}{{{b_3} - {b_2}}},\frac{1}{{{b_4} - {b_3}}},\frac{1}{{{b_5} - {b_4}}}$ are in an A.P. with common difference –2) is correct because by evaluating the integral and finding the differences between consecutive terms, we can show that the reciprocals form an arithmetic progression.

Option D ((\log x)^-1 · x(x-1)) is correct because it is the result of applying Leibniz's rule for differentiating an integral with variable limits.

Option B (2) is correct because the integral simplifies to the integral of |sin x| over the interval [-$\pi/2$, $\pi/2$], which evaluates to 2.

Option A ($\frac{\pi }{4}$) is correct because by using the property of definite integrals and adding the two forms of the integral, we can simplify the expression and evaluate it.

Option C ($\mathrm{2}$) is correct because the integral of the absolute value function is evaluated by recognizing that |1-x| = 1-x on the interval [-1, 1].

Option C, $\log \left( {4/e} \right)$, is the correct area calculated by integrating the function y = log x from x = 1 to x = 2 and simplifying the result.

Option A (16/15) is correct because the integral of sin⁵(x/2) from 0 to pi evaluates to 16/15.