Definite integration practice · Mixed difficulty

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 D (-π/2) is correct because using the property ∫0π[cot x] dx = ∫0π[-cot x] dx and [x] + [-x] = -1, the integral simplifies to -π/2.

Option C (4) is correct because differentiating F(x²) and solving for f(x) allows us to find f(4) = 4.

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 C (0) because the integral simplifies to the integral of an even function plus the integral of an odd function, and the integral of the odd function over the symmetric interval is zero.

Option B ($2 \alpha$) is correct because using the property $\int_0^a f(x) dx = \int_0^{a/2} [f(x) + f(a-x)] dx$ and the given condition $f(a-x)=f(x)$, we get $\int_0^a f(x) dx = 2 \int_0^{a/2} f(x) dx = 2\alpha$.

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

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 B ($a = 2,\,b = 14,\,n \in R$) is correct because it satisfies the property of definite integrals where the integral's value depends on the limits of integration, and $a+b = 16$ and $\frac{b-a}{2} = 6$.

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

Option A ($\frac{A}{2}$) is incorrect. The correct answer is B (A). The provided solution incorrectly states the answer is A/2. The correct area is A.

Option A is correct because the integral of [x] from -1/2 to 1/2 is -1/2, and the integral of the log term is zero due to it being an odd function.

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

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 A is correct because by substituting $x \tan \theta = z \sin \theta$ and changing the limits of integration, the integral can be simplified to the form given in option A.

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 ($\frac{a+b}{2} \int\limits_{{a}}^{b} f(x) d x$) is correct because using the property $f(a+b-x) = f(x)$ simplifies the integral to this form.

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 A ($\frac{1}{\sqrt{2}} \log |\tan (\frac{x}{2}+\frac{3 \pi}{8})|+C$) is correct because the integral simplifies to a secant integral, which can be solved using the tangent half-angle formula.

Option B ($\log(\frac{e}{2})$) is correct because it represents the area bounded by the inverse function, X-axis, and the given limits.

Option C ($\pi^2$) is correct because the integral can be simplified using the symmetry of the absolute value function.

Option A (-11/2) is correct because the definite integral is evaluated by splitting the integral based on the greatest integer function and absolute value.