calculus please solve questionnnn 4 **Sample Mathematics Exam Questions for Advanced Calculus**1. **Evaluate the logarithmic expression:** \[ \log_6(9) + \log_6(4) \] - A) \(\log_6(13)\) - B) \(2\) - C) \(\log_6\left(\dfrac{9}{4}\right)\) - D) \(6\) 2. **Solve the equation:** \[ \ln\left(\dfrac{x-1}{x+1}\right) = 2 \] - A) \(\dfrac{e^2+1}{e^2-1}\) - B) \(\dfrac{e^2+1}{1-e^2}\) - C) \(\dfrac{2+2e}{1-e}\) - D) \(\dfrac{e^2-1}{e^2+1}\) 3. **Find the solutions of the quadratic equation:** \[ e^{2x} + e^x - 6 = 0 \] - A) \(x = \ln(2), \ln(3)\) - B) \(x = \ln(2), \ln(-3)\) - C) \(x = 2, -3\) - D) \(x = \ln(2)\) 4. **Assume \( f \) is a differentiable function, then evaluate the integral:** \[ \int x^3 f''(x) \, dx \] - A) \(x^3 f(x) - \dfrac{1}{4} x^4 f(x) \, dx\) - B) \(\dfrac{1}{4} x^4 f(x) + c\) - C) \(x^3 f''(x) - 3 \int x^2 f'(x) \, dx\) - D) \(x^3 f(x) - 3 \int x^2 f(x) \, dx\) 5. **Which of the following improper integrals converge?** - A) \(\int_0^\infty x^{-8} \, dx\

As you mentioned for only Question number 4,I do solve only Ques 4 as per your mentioned details.

(1) log (9)+ log, (4) is equal to: B) 2 A) logo (13) (2) Solve the equation In (1) = 2: e²+1 e²+1 A) 2+1 e2-1 B) C) 1-82 (3) The solutions of e2x + ex-6=0 are: A) x = ln(2), In(3) A) (6) Suppose that x = 3 tan 8, then cos 0 = =: √x²-9 A) √√9-x² B) C) √9+x² x dx = C) x = 2,-3 D) x = In (2) (4) Assume f is a differentiable function, f x³ f'(x) dx = A) x³ f(x) - fx¹ f(x) dx B) x = ln(2), In(-3) B)x f(x) + c C) x³ f"(x)-3 f x² f(x) dx (5) Which of the following improper integrals converge: A) x dx B)e-2x dx C) logo (2) B) 4 (9) The derivative of 2 sin (2sin-¹x) In 2 A) √1-x2 2+2e -1 (7) According to the method of partial fractions, C) -1 B) A) -2 -8 4 (8) √3 x²+9 TT -1x C) is: 2sin 1x √1-x² B) D) D) 2+2e 1-e C) D) 6 √x²+9 3x (x-1)(x-2)(x-3) D)3 D) Diverges In 2 (2sin-1x) In √1+x² = D) x³ f(x) - 3 f x² f(x) dx D) In (x) dx (x-1) + B (x-2) + (x-3) the value of B is D) (2 cos-¹x) In 2 √1-x²