QUESTION 1a)

Find the area enclosed between the two curves with equations 9 X= and X = 49 Area enclosed between the curves isUse the method of partiai fractions to express 2x2x+l x3x a as the sum of 3 terms of the form , where a, m and mx+c c are constants. You will need to solve a system of 3 linear equations to find the coefficients. 2x2x+l 3: + XX Here we will use a u subtitution to find the antideriyatiye / sin2(x) c037 (x)dx Note: You will not be able to progress through the problem until you have correctly answered previous part. This will require you to submit answers for each part. Extra attempts have been allocated accordingly. Part 1: Identify u /\\ The most appropriate choice of u is: sin(x) 3 Part 2: Substitution part 1 /\\ do Express the integrand in terms of u and : x (in sin2(x) COST (3:) = d x 2. In this assignment you will evaluate the integral ff(x)dx where 205 - 24 - 14x2 + 4x - 8 f (a) = 23 - 22 - x - 15 (i) Using long division show that 2x2 + 5x +7 f (20) = 202+1+ 203 - 202 - x - 15(ii) Let q(:r:) = 3:3 3:2 rs 15. Check that (1(3) = 0. This means that :L' 3 is a divisor of (1(3). To verify this, use long division to nd constants a, b, C E R such that q(:r:) = (.7; 3)(a$2 + by; + c) (iii) Explain why the ax2 + ba + c you found in (ii) is irreducible. (iv) Set p(x) = 2x2 + 5x + 7. Using the method of partial fractions explained in lectures, and the factorisation of q(x) in (ii) write p(a) A Bx + C -+ x - 3 ax2 + bac + c for some constants A, B, C ER.(v) Using the above evaluate the integral Jf(x)dx where (f(x)da = (x2 + 1)da +/ A Bx+ C dx + dx . x - 3 ax2 + bx + c