Exercise 10: Let fn be a function such that f," (x) = fn(x) (with n € N). Let g be any sufficiently continuously differentiable function (specifically, g should be n-times continuously differentiable). (a) Use integration by parts to show that: | f3(x)(g(x) + g" (x)) dr = f"(x)g(x) – f%(x)g'(x)+ f3(x)g"(x) + C. Hint: Apply integration by parts to S f(x)g(x) dx. (b) Use the pattern observed to guess a similar formula for fn, with n e N. (c) Check your formula, using integration by parts, on f2(x) = cosh(x). Possibly correct your guess from (b). (d) Use this formula to compute | cos(x) (210 – 50402°°) dx. (e) Use this formula to compute cos(x)x10 dx. Hint: (d1²/dx12) cos(x) = cos(x). (f) What does the formula in the n = 1 case say? Use this to compute: | e* sec(x)(1+ tan(x)) dx. Note that neither summand is, by itself, easy to integrate.

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Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.