Definite integral

1003124305

Level: 
C
Given a function \( f(x)=ax^6+bx^3+cx+8 \), find real numbers \( a \), \( b \) and \( c \), such that \( \int\limits_0^1f(x)\,\mathrm{d}x=\frac{35}4 \), \( f'(0)=2 \) and \( f'(1)=180 \).
\( a=7 \), \( b=-5 \), \( c=2 \)
\( a=7 \), \( b=5 \), \( c=2 \)
\( a=-7 \), \( b=-5 \), \( c=2 \)
\( a=-7 \), \( b=5 \), \( c=-2 \)

1003124303

Level: 
C
Which of the given values of real numbers \( a \), \( b\in\left(0;\frac{\pi}2\right) \), such as \( a < b \), makes the equality \( \int\limits_a^b \cos x\,\mathrm{d}x=2\cos\frac{\pi}4\cdot\sin\frac{\pi}{12} \) true?
\( a=\frac{\pi}6 \), \( b=\frac{\pi}3 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}6 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}4 \)
\( a=\frac{\pi}4 \), \( b=\frac{\pi}3 \)

1103124301

Level: 
C
The picture shows graphs of two quadratic functions \( f_1(x) \) and \( f_2(x) \). Find the unknown real positive constant \( a \) (as shown in the picture) such that the value of the definite integral \( \int\limits_{-1}^1 f_1(x)\,\mathrm{d}x \) is greater by \( 8 \) than the value of the definite integral \( \int\limits_{-1}^1 f_2(x)\,\mathrm{d}x \).
\( a = 3 \)
\( a = 1 \)
\( a = 4 \)
\( a = 6 \)