Question
(a) Find
(i)\(\int \frac{e^{2x}+6}{e^{2x}}\),
(ii) \(\int 3\cos ^{2}xdx\),
(b) Use the trapezium rule with 2 intervals to estimate the value of,
\(\int _{1}^{2}\frac{6}{\ln (x+2)}dx\),
giving your answer correct to 2 decimal places.
Answer/Explanation
(a) (i) Attempt to divide by \(e^{2x}\) and attempt to integrate 2 terms
Integrate a term of form \(ke^{-2x}\) correctly.
Fully correct integral \(x-3e^{-2x}\).
(ii) State correct expression \(\frac{1}{2}\cos 2x+\frac{1}{2}\) or equivalent
Integrate an expression of the form a + b cos 2x, where ab ≠ 0, correctly
State correct integral \(\frac{3\sin 2x}{4}+\frac{3x}{2}\).
(b) State or imply correct ordinates 5.46143…, 4.78941…, 4.32808…
Use correct formula, or equivalent, correctly with h = 0.5 and three ordinates
Obtain answer 4.84 with no errors seen