Antiderivative Of Secx Antiderivative Of Secx
We have to find the Antiderivative of secx tanx. To find the antiderivative of secx tanx we must first understand the meaning or definition of antiderivative. Antiderivative is an operation which is opposite of derivation operation that means antiderivative calculates the integral of a derivative. Suppose we have a function f(x), its derivative is g(x) means d(f(x))= g(x) than antiderivative of g(x) is f(x) that is ∫ g(x) dx = f(x) + c dx. So we get f(x) as antiderivative of g(x) is a constant. Proof. It is not possible to prove that by applying the usual theorems on limits (Lesson 2). We have to go to geometry, and to the meanings of sin θ and radian measure. Let O be the center of a unit circle, that is, a circle of radius 1; and let θ be the first quadrant central angle BOA, measured in radians. Then, since arc length s = rθ, and r = 1, arc BA is equal to θ. (Topic 14 of Trigonometry.) Know More About Inverse Functions Worksheet
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Draw angle B'OA equal to angle θ, thus making arc AB' equal to arc BA; draw the straight line BB', cutting AO at P; and draw the straight lines BC, B'C tangent to the circle. Then BB' < arc BAB' < BC + CB'. Now, in that unit circle, BP = PB' = sin θ, (Topic 17 of Trigonometry), so that BB' = 2 sin θ; and BC = CB' = tan θ. (For, tan θ = BC OB = BC 1 = BC.) The continued inequality above therefore becomes: 2 sin θ < 2θ < 2 tan θ. On dividing each term by 2 sin θ: 1< θ sin θ < 1 cos θ . (Problem 2.) And on taking reciprocals, thus changing the sense: 1> sin θ θ > cos θ. (Lesson 11 of Algebra, Theorem 5.) On changing the signs, the sense changes again : −1 < − sin θ θ < −cos θ, (Lesson 11 of Algebra, Theorem 4), and if we add 1 to each term: 0< 1− sin θ θ < 1 − cos θ. Read More About Calculus Worksheet
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Now, as θ becomes very close to 0 (θ 0), cos θ becomes very close to 1; therefore, 1 − cos θ becomes very close to 0. The expression in the middle, being less than 1 − cos θ, becomes even closer to 0 (and on the left is bounded by 0), therefore the expression in the middle will definitely approach 0. This means: Here we have used method of Integration by parts to calculate the antiderivative of sec x tan x. Hence to find the antiderivative of secxtanx we have to calculate the integral of secxtanx which is sec (x) + C where as C is any arbitary constant generated after integration.
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