Definite integral

1003108203

Level: 
B
Compare the definite integral \( I=\int\limits_0^{\frac{\pi}4}\frac{\cos⁡2b}{\cos^2⁡b}\,\mathrm{d}b \) to the number \( \frac{\pi}2 \).
\( I \) is smaller than \( \frac{\pi}2 \) by \( 1 \).
\( I \) is bigger than \( \frac{\pi}2 \) by \( 1 \).
\( I \) is equal to \( \frac{\pi}2 \).
\( I \) is smaller than \( \frac{\pi}2 \) by \( \frac{\pi}4 \).

1003108205

Level: 
B
Compare the two definite integrals \( I_1=\int\limits_{-1}^1\left(x+\frac{\pi}2\right)\mathrm{d}x \) and \( I_2=\int\limits_0^{\frac{\pi}4}\mathrm{tg}\,x\cdot\cos ⁡x\,\mathrm{d}x \).
\( I_1 \) is bigger than \( I_2 \).
\( I_1 \) is smaller than \( I_2 \).
\( I_1 \) is equal to \( I_2 \).
These integrals cannot be compared.

1003118801

Level: 
B
Which of the following formulas is not equal to \( \int\limits_4^8\frac{3x+1}{x^2-x-6}\,\mathrm{d}x \)?
\( \int\limits_4^8\frac2{x-3}\,\mathrm{d}x -\int\limits_4^8\frac1{x+2}\,\mathrm{d}x \)
\( \int\limits_4^8\frac2{x-3}\,\mathrm{d}x + \int\limits_4^8\frac1{x+2}\,\mathrm{d}x \)
\( \int\limits_4^6\frac{3x+1}{x^2-x-6}\,\mathrm{d}x + \int\limits_6^8\frac{3x+1}{x^2-6-x}\,\mathrm{d}x \)
\( \int\limits_8^4\frac{-1-3x}{x^2-x-6}\,\mathrm{d}x \)

1003118804

Level: 
B
Evaluate the integral \( \int\limits_{-3}^2 f(x)\,\mathrm{d}x \), where \( f(x)=x^{-4} \) for \( x\in\left[-3;-\frac12\right] \) and \( f(x)=12.8-6.4x \) for \( x\in\left[ -\frac12;2\right]\). (rounded to \( 2 \) decimal places)
\( 22.65 \)
\( 43.\overline{8} \)
\( 44.\overline{1} \)
\( 29.7\overline{1} \)