4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0
4. (a Let (sin( x cos( ) dr...
W60. Compute (sin x + cos x)(4 – 2 sin 2x – sin? 2x)e" dx sin 2x where = 6 (0,7) Mihály Be Pirkulyiev Rovsen W38. Let (an)n be a sequence, given by the recurrence: man+1 + (m - 2) an an-1 = 0 where me R is a parameter and the first two terms of (an), are fixed known real numbers. Find me R, so that lim an = 0 n-00 Tad
Evaluate the following integral. 1/2 7 sin ?x -dx 1 + cos x 0 1/2 7 sin 2x dx = V1 + cos x 0 Score: 0 of 1 pt 1 of 10 (0 complete) HW Score: 0%, 0 of 10 pts 8.7.1 A Question Help The integral in this exercise converges. Evaluate the integral without using a table. dx x +49 0 dx X2 +49 (Type an exact answer, using a as needed.) 0
15. Using that sin' (2) = cos(x), cos' (2) = - sin() show that arccot (0) = 1 +22
4. Integration: TT (a) Si Jachtvoz dx (b) sin x dx (6-cos x)3 (c) , (2-2)' (3) dx (d) St(t - 5)8dt 6x7-x*+VX-4 dx x2
If sin(x) = and cos(x) < 0 and 0 (5) and sin⑨ x < 2T. determine the exact value of cos
If sin(x) = and cos(x)
Use the product rule to differentiate 4 sin x cos x. dx
(1 point) Solve the following differential equation: (tan(x) 8 sin(x) sin(y))dx + 8 cos(2) cos(y)dy = 0. = constant. help (formulas)
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but we would be left with an expression in terms of sin(7x) with no extra cos(7x) factor. Instead, we separate a single sine factor and rewrite the remaining sin" (7x) factor in terms of cos(7x): sin'(7x) cos”(7x) = (sinº(7x))2 cos(7x) sin(7x) = (1 - Cos?(7x))2 cos?(7x) sin(7x). in (7x) cos?(7x) and ich is which? Substituting u = cos(7x), we have du = -sin (3x) X...
13. Evaluate, S sin 5x cos x dx. Also prove that, 52" sin mx cos nx dx = 0 Using reduction formula *****