Definite integration practice · Mixed difficulty

Option C (4) is correct because the integral simplifies to the integral of an odd function plus a constant, and the integral of the odd function over a symmetric interval is zero.

Option A (2) is correct because by using the property of definite integrals and the given function, the ratio simplifies to 2.

Option D ($\pi$) is correct because the integral simplifies to a constant value after applying trigonometric identities and properties of definite integrals.

Option D (All of these) is correct because the integral I is bounded by $\frac{1}{2}$ and $\frac{\pi}{6}$, and it is also greater than 0.

The correct answer is B (100). The integral of sin(x) from -100 to 100 involves using properties of definite integrals and recognizing that the integral of sin(x) is -cos(x).

Option C ($\frac{4}{3}$) is correct because the integral of |sin³θ| from 0 to π is equal to the integral of sin³θ from 0 to π, which evaluates to 4/3.

The correct answer is C (1/2) because the integrand is the derivative of a simple trigonometric function, allowing for direct evaluation using the fundamental theorem of calculus.

Option C ($\frac{{{\pi ^2}}}{{2\sqrt 2 }}$) is correct because it is the result of evaluating the definite integral using properties of definite integrals and trigonometric substitutions.

Option B ($e-1$) is correct because the limit of the Riemann sum is evaluated as a definite integral from 0 to 1 of $e^x$.

Option A ($\frac{1}{4} \ln \frac{3}{2}$) is correct because after substitution and using the property $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$, the integral simplifies to $\frac{1}{4} \ln \frac{3}{2}$.

Option D (-1) is correct because the integral evaluates to (π/2) - 1, which can be expressed as (1/2)(π - 2). Therefore, m = 1/2 and n = -2, and m*n = -1.

Option B ($\frac{{1}}{{15}}\,{\log _e}\,6$) is correct because it is the result of evaluating the definite integral using trigonometric substitution and logarithmic properties.

Option D, $\mathrm{4\;sq\;units\;}$, is the correct area because the equation represents an ellipse, and its area is calculated using the formula $\pi ab$, where a and b are the semi-major and semi-minor axes.

Option C ($\pi /2$) is correct because it is the value obtained after evaluating the definite integral using the property of definite integrals.

Option D (2) is correct because the integral of the absolute value of cos(x) from 0 to pi is calculated by splitting the integral into two parts, from 0 to pi/2 and from pi/2 to pi, and then evaluating each part separately.

Option C ($2 \pi$) is correct because after expanding the exponential and integrating term by term, only the constant term contributes to the integral.

Option C is incorrect. Option B is correct because the definite integral of an odd function over a symmetric interval is zero, leaving only the integral of the constant term, which depends on 'c'.

Option D is correct because the integral of the given quadratic function can be expressed in terms of the function's values at specific points, matching Simpson's 1/3 rule.

Option D (sin t - cos t + 4n + 1) is correct because the integral is evaluated by recognizing the periodicity of the integrand and using properties of definite integrals.

Option A is correct because by splitting the integral into intervals where [x] is constant and using integration by parts, we can derive the given expression.

Option D is correct because the function f(t) is found by differentiating the integral equation and then substituting the value to find f(1/$\sqrt{3}$).

Option D (1) is correct because the integral can be solved by partial fraction decomposition and evaluating the resulting logarithmic terms.

Option B ($\frac{1}{{\left( {n + 1} \right)\,\,\left( {n + 2} \right)}}$) is correct because after applying the property of definite integrals and simplifying, we arrive at this result.

Option D ($\int\limits_{x + 5}^5 {g(t)\,dt}$) is correct because after applying the properties of even and odd functions, and using substitution, the integral simplifies to this form.

Option C ($10\\,\,\left( {\frac{\pi }{2}} \right)^9$) is correct because by applying integration by parts twice, a recursive relationship between u_10 and u_8 is established, leading to the given result.

Option A (1) is correct because the integral of the fractional part function over the interval [-1, 1] can be calculated by splitting the integral and using the definition of the fractional part function.

Option B ($\sqrt 2 - 1$) is correct because it represents the definite integral of the difference between $\cos x$ and $\sin x$ from 0 to $\frac{\pi}{4}$.

Option B (4) is correct because the integral of the absolute value function is evaluated by splitting the integral into intervals where the function inside the absolute value is positive or negative.

Option B is correct because the integral can be simplified using the periodic property of the function, leading to the result 200/$\pi$.

Option A (1) is correct because the integral can be simplified by using the property of definite integrals and substituting x = -t.