Definite integration practice · Easy

Option B is correct because using the property of definite integrals $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$, we can simplify the given integral and find its value.

Option A ($\frac{2}{5}$) is correct because differentiating the given integral and solving for $f(t^2)$ yields $f(\frac{4}{25}) = \frac{2}{5}$.

Option B ($\ - \sqrt 2 - \sqrt 3 - 1$) is correct because evaluating the definite integral of the greatest integer function results in this value.

Option B ($\frac{{\pi}}{{4}}$) is correct because it is the result of using the trigonometric identity to express $\sin^3 x$ and then applying the given integral value.

Option C (H.P.) is correct because the reciprocals of the given terms form an arithmetic progression.

The correct answer is D (${(105)^2} \times {2^{36}}$) after evaluating the determinant of the adjoint of the inverse of matrix A, which involves integral calculations and properties of determinants.

Option B ($\frac{\pi}{4}$) is correct because using the property $\int_0^{a} f(x) dx = \int_0^{a} f(a-x) dx$ simplifies the integral to $\frac{\pi}{4}$.

Option D ($\mathrm{4}$) is correct because the integral of the absolute value function is calculated by splitting the integral at the point where the function changes sign.

Option D ($\frac{\pi}{2}$) is correct because solving the integral and the resulting trigonometric equation leads to $\theta = \frac{\pi}{2}$ within the given range.

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 C ($\frac{{7}}{{3}}$ sq. unit) is correct because the area is calculated by integrating $x$ with respect to $y$ from 1 to 4 after expressing $x$ in terms of $y$.

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

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).

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 D ($\frac{x}{\sqrt{1+x^2}}$) is correct because the derivative of $\sqrt{b^2+1}$ with respect to $b$ gives the function $f(x)$.

Option A (2n/3) is correct because the integral evaluates to 2n/3 after summing the integrals over the intervals [m-1, m].

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 A (2) is correct because the integral from -2 to 2 of the odd function $e^{\cos x} \cdot \sin x$ is 0, and the integral from 2 to 3 of 2 is 2.

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 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 (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 A, 0, is correct because the limit evaluates to e - 2, and the greatest integer function of e - 2 is 0 since 2 < e < 3.

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 ($\frac{\pi}{3}$) is correct because after solving the trigonometric equation and finding the limits, the definite integral evaluates to $\frac{\pi}{3}$.