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Q4. (a) Use the Fundamental Theorem of Calculus to find the derivative with respect to \(x\) of the function \(\int_2^{\ln x} \sin^2(t) dt\). (b) Evaluate the following three integrals, showing your work clearly in each case: (i) \(\int \frac{x^2 - 8x - 6}{(x^2 + 2)(x - 2)} dx\) (ii) \(\int_0^{\pi/2} x \cos x \, dx\) (Hint: use integration by parts with \(u = \cos x\).) (iii) \(\int_0^{\infty} \frac{x}{1 + x^4} dx\). (Hint: consider \(u = x^2\).) Q5. (a) Compute the area of the finite region bounded by the curves \(f(x) = x^2 - 4x\) and \(g(x) = 2x - 5\). (b) Calculate the length of the curve given by \(h(x) = \frac{2}{3}(x - 1)^{3/2}\) from \(x = 1\) to \(x = 4\). (c) Find the area of the surface formed by rotating the curve \(y = \frac{x^3}{9}\) between \(x = 0\) and \(x = 2\) about the \(x\)-axis.

          Q4. (a) Use the Fundamental Theorem of Calculus to find the derivative with respect to \(x\) of the function \(\int_2^{\ln x} \sin^2(t) dt\).
(b) Evaluate the following three integrals, showing your work clearly in each case:
(i) \(\int \frac{x^2 - 8x - 6}{(x^2 + 2)(x - 2)} dx\)
(ii) \(\int_0^{\pi/2} x \cos x \, dx\) (Hint: use integration by parts with \(u = \cos x\).)
(iii) \(\int_0^{\infty} \frac{x}{1 + x^4} dx\). (Hint: consider \(u = x^2\).)
Q5. (a) Compute the area of the finite region bounded by the curves \(f(x) = x^2 - 4x\) and \(g(x) = 2x - 5\).
(b) Calculate the length of the curve given by
\(h(x) = \frac{2}{3}(x - 1)^{3/2}\)
from \(x = 1\) to \(x = 4\).
(c) Find the area of the surface formed by rotating the curve \(y = \frac{x^3}{9}\) between \(x = 0\) and \(x = 2\) about the \(x\)-axis.

Q4. (a) Use the Fundamental Theorem of Calculus to find the derivative with respect to x of the function ∫2^ln xsin^2(t) dt.
(b) Evaluate the following three integrals, showing your work clearly in each case:
(i) ∫(x^2 - 8x - 6)/((x^2 + 2)(x - 2)) dx
(ii) ∫0^π/2 x cos x dx (Hint: use integration by parts with u = cos x.)
(iii) ∫0^∞(x)/(1 + x^4) dx. (Hint: consider u = x^2.)
Q5. (a) Compute the area of the finite region bounded by the curves f(x) = x^2 - 4x and g(x) = 2x - 5.
(b) Calculate the length of the curve given by
h(x) = (2)/(3)(x - 1)^3/2
from x = 1 to x = 4.
(c) Find the area of the surface formed by rotating the curve y = (x^3)/(9) between x = 0 and x = 2 about the x-axis.

Added by Ashley C.

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