Definite integration practice · Easy

Option D (0) is correct because integration by parts simplifies the integral to zero, utilizing the given boundary conditions.

Option B ($\frac{4}{3}$) is correct because the integral simplifies to an integral of sin(x)sqrt(cos(x)) which can be solved by substitution.

Option C ($2 c \log 2$ sq. unit) is correct because the area is calculated by integrating $y = \frac{{c}}{{x}}$ from $x=1$ to $x=4$, which simplifies to $c \log 4 = 2c \log 2$.

Option B ($\frac{5}{2}$) is correct because the integral is evaluated by splitting it into two parts based on the sign of the logarithm and then integrating.

Option D (28/3) is correct because the integral of the absolute value function |1-x²| is computed by splitting the integral into intervals where the function inside the absolute value is positive or negative, and then integrating each part separately.

Option D ($(1 / 8) \log 3$) is correct because it is the result of evaluating the definite integral after a suitable substitution.

Option C ($\frac{1}{n+1}-\frac{1}{n+2}$) is correct because it is the result of integrating by parts and using the properties of definite integrals.

Option A (less than $10^{-7}$) is correct because the absolute value of the integral is bounded by the integral of the absolute value of the integrand, which is less than $10^{-7}$.

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

Option D ($\frac{1}{2}\,\frac{{\sqrt {\pi } }}{a}$) is correct because it follows from a u-substitution and the given integral value.

Option A ($\log \,\,\left( {\frac{b}{a}} \right)$) is correct because the limit of the Riemann sum is correctly converted to a definite integral and evaluated.