Definite integration practice · Easy

Option B ($\log \left(10^x+x^{10}\right)+c$) is correct because the integral is in the form of $\int \frac{{f'(x)}}{{f(x)}} dx = \log |f(x)| + c$, where $f(x) = 10^x + x^{10}$.

Option A is correct because the integral is solved by using a substitution to transform it into a standard form that can be easily integrated.

Option D ($\frac{\mathrm{\pi}}{4}$) is correct because the integral can be solved using properties of definite integrals and trigonometric identities.

Option A (1) is correct because the integral of x from -1 to 1 is 0, and the integral of -[2x] from -1 to 1 is -(-2 - 1 + 0 + 1)/2 = 1.

The correct answer is A (1) because the integral of the absolute value function |x-1| from 0 to 2 is equal to 1.

Option D (none of these) is correct because the value of the integral is 0, which is not among the provided options.

The correct answer is -$\sqrt{2}$. The integral is solved using the inverse secant function, and solving for x gives -$\sqrt{2}$.

Option C is correct because the integral simplifies to $\int_0^{400\pi} \sqrt{2}|\sin x| dx$, and since $|\sin x|$ is periodic with period $\pi$, the integral becomes $400\sqrt{2}\int_0^{\pi} \sin x dx = 800\sqrt{2}$.

Option B is correct because the integral from 0 to 2a can be split into two integrals, each from 0 to a, and then using the given information, the result is $\lambda + \mu$.

Option C is correct. The integral is evaluated by splitting it into two parts based on the greatest integer function.

Option B (2) is correct because it is the value obtained after evaluating the definite integral of the absolute value function.

Option B ($2 \sqrt{2}(\sqrt{2}-1)$) is correct because it represents the area calculated by integrating the difference between the two functions over the given interval, considering the absolute value.

Option A (| b\ | - | a\ |$\left| {\,b\} \right|\, - \,\left| {\,a\} \right|$) is correct because the integral of |x|/x from a to b, where a < 0 < b, simplifies to b - (-a) = |b| - |a|.

Option B (3/2) is correct because the integral of the max function involving fractional parts is evaluated piecewise and summed.

Option C (2) is correct because the integral simplifies to the integral of (sin x + cos x) from 0 to pi/2, which evaluates to 2.

Option A ($\frac{13}{\sqrt{165}}$) is correct because it represents the shortest distance between the two given lines, calculated using the formula involving the cross product of direction vectors and the difference of position vectors.

Option C ($1000(e-1)$) is correct because the integral simplifies to a sum of identical integrals due to the periodicity of the fractional part function.

Option A (1) is correct because $\int_{-a}^{a} f(x) dx = \int_{0}^{a} (f(x) + f(-x)) dx$, so $K=1$.

Option C (${x^x} + \,c$) is correct because the integral of ${x^x}\,(1 + \log \,|{\,x\}|\,)\,dx}$ is ${x^x} + \,c$ after applying substitution.

Option B ($5:8$) is correct because the area under the curve is calculated using integration and then compared to the area of a circle with radius $a$.

Option C ($-\frac{1}{2}$) is correct because it is the value of the function f(x) after differentiating the given integral equation and substituting $x = \frac{\pi}{6}$.

Option A ($\frac{3}{8}$) is correct because it is the value obtained after evaluating the definite integral using trigonometric substitution.

Option A ($\frac{{4}}{{3}}$) is correct because the integral simplifies after factoring and using the even function property of cosine.

Option C ($\int_0^1\left(1-\sqrt{1-y^2}\right) d y$) is the correct answer because it results from simplifying and rearranging the given integral equation.

Option A ($\frac{2}{5}$) is correct because differentiating both sides and substituting $t = \frac{2}{5}$ gives $f(\frac{4}{25}) = \frac{2}{5}$.

Option C is correct because the integral of f(t) from 0 to x is calculated piecewise, resulting in the given expression for x in [2,3].

Option A ((9x^2-4x)log x) is correct because it is the result of applying Leibniz's rule for differentiating an integral with variable limits.

Option D ($\log_e \frac{{4}}{{3}}$) is correct because it is derived by substituting $t = \tan x$ and evaluating the definite integral.

Option B ($5-\sqrt{3}-\sqrt{2}$) is correct because the integral of the greatest integer function of x squared from 0 to 2 is equal to $5-\sqrt{3}-\sqrt{2}$.

Option B (7/3) is correct because the integral is evaluated by splitting the interval based on the greatest integer function's value.