Fundamental theorem; properties of definite integrals — practice questions

Question 1 · difficulty L2 · understanding

The value of $$12\int\limits_0^3 {\left| {{x^2} - 3x + 2} \right|dx}$$ is ____________

Question 2 · difficulty L2 · understanding

The value of the integral $$\int\limits_0^\pi {|{{\sin }\,}2x|dx}$$ is ___________.

Question 3 · difficulty L2 · understanding

Let $f$ be a polynomial function such that $f\left(x^2+1\right)=x^4+5 x^2+2$, for all $x \in \mathbb{R}$. Then $\int\limits_0^3 f(x) d x$ is equal to

• A. $\frac{33}{2}$
• B. $\frac{5}{3}$
• C. $\frac{27}{2}$
• D. $\frac{41}{3}$

Question 4 · difficulty L2 · understanding

The integral \int_\limits{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x is equal to

• A. $$-1/2$$
• B. $$-1/4$$
• C. 1/4
• D. 1/2

Question 5 · difficulty L2 · understanding

The value of the integral $$\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx}$$ is :

• A. 2
• B. 0
• C. $$-$$1
• D. 1

Question 6 · difficulty L2 · understanding

Which of the following statements is correct for the function g($$\alpha$$) for $$\alpha$$ $$\in$$ R such that $$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx}$$

• A. $$g(\alpha )$$ is a strictly increasing function
• B. $$g(\alpha )$$ is an even function
• C. $$g(\alpha )$$ has an inflection point at $$\alpha$$ = $$-$$$${1 \over 2}$$
• D. $$g(\alpha )$$ is a strictly decreasing function

Question 7 · difficulty L2 · understanding

The value of the definite integral $$\int\limits_{0}^{2} \frac{1}{3^x + 3} dx$$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{\log_e 3}{3}$
• D. $\frac{\log_e 3}{2}$

Question 8 · difficulty L3 · understanding

If $\alpha = \int\limits_{0}^{2\sqrt{3}} \log_{2}(x^{2} + 4) \, dx + \int\limits_{2}^{4} \sqrt{2x - 4} \, dx$, then $\alpha^{2}$ is equal to ________.

Question 9 · difficulty L3 · understanding

If $f(x)$ satisfies the relation $f(x)=e^x+\int_0^1\left(y+x e^x\right) f(y) d y$, then $e+f(0)$ is equal to $\_\_\_\_$ .

Question 10 · difficulty L3 · understanding

Let a differentiable function $f$ satisfy the equation $\int_0^{36} f\left(\frac{t x}{36}\right) d t=4 \alpha f(x)$. If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4, \beta)$, then $\beta^\alpha$ is equal to $\_\_\_\_$ .