علمِ احصا

Written by Hassan Raza (Mathematics)

نیٹ · ریاضی · 10 سوالات

نام: ____________________ تاریخ: ____________

What is d/dx of x² ln x?
- A 2x / x
- B 2x ln x + x
- C x ln x
- D 2x ln x
What is the integral of 1/x dx for x > 0?
- A x ln x + C
- B ln x + C
- C -1/x² + C
- D 1/x² + C
What is the value of the integral of sin x from 0 to pi?
- A pi
- B 0
- C 1
- D 2
What is the limit of (3x² + 1) / (x² - 5) as x tends to infinity?
- A 0
- B 3
- C Infinity
- D -1/5
What is d/dx of tan x?
- A -cosec² x
- B cot x
- C sec² x
- D sec x tan x
What is the maximum value of f(x) = 3 + 4x - x²?
- A 3
- B 7
- C 2
- D 4
What is the area under the line y = x from x = 0 to x = 4?
- A 8
- B 4
- C 16
- D 12
What is d/dx of sin(x²)?
- A 2 sin x cos x
- B 2x cos(x²)
- C cos(x²)
- D -2x cos(x²)
What is the integral of x e^x dx?
- A e^x (x + 1) + C
- B x² e^x / 2 + C
- C x e^x + C
- D e^x (x - 1) + C
What is d/dx of sqrt(x² + 1)?
- A sqrt(x² + 1) / x
- B 2x sqrt(x² + 1)
- C x / sqrt(x² + 1)
- D 1 / (2 sqrt(x² + 1))

جوابی کلید

1. B, 2. B, 3. D, 4. B, 5. C, 6. B, 7. A, 8. B, 9. D, 10. C

By the product rule, the derivative is 2x ln x + x² x (1/x) = 2x ln x + x.
The derivative of ln x is 1/x, so the integral of 1/x is ln x + C.
The antiderivative is -cos x. So the value is -cos(pi) + cos(0) = 1 + 1 = 2.
Divide top and bottom by x². The terms with x in the denominator vanish, leaving 3 / 1 = 3.
The derivative of tan x is sec² x.
f'(x) = 4 - 2x = 0 gives x = 2, and f(2) = 3 + 8 - 4 = 7. The parabola opens downward, so this is a maximum.
The area is the integral of x from 0 to 4, which is 16/2 = 8. It is also the area of a triangle with base 4 and height 4.
By the chain rule, the derivative is cos(x²) multiplied by the derivative of x², which is 2x.
Integration by parts gives x e^x - e^x + C, which equals e^x (x - 1) + C.
By the chain rule, the derivative is 1/2 (x² + 1)^(-1/2) x 2x = x / sqrt(x² + 1).