Contents
What are the Antiderivatives of trig functions?
This Section: 4. Integrals of Trigonometric Functions
| Derivative Rule | Antiderivative Rule |
|---|---|
| d dx sin x = cos x | cos x dx = sin x + C |
| d dx cos x = − sin x | sin x dx = − cos x + C |
| d dx tan x = sec2x | sec2x dx = tan x + C |
| d dx cotan x = − cosec2x | cosec2x dx = − cotan x + C |
What is derivative of basic trigonometric functions?
In trigonometry, differentiation of trigonometric functions is a mathematical process of determining the rate of change of the trigonometric functions with respect to the variable angle. Derivation of sin x: (sin x)’ = cos x. Derivative of cos x: (cos x)’ = -sin x. Derivative of tan x: (tan x)’ = sec2 x.
What is the difference between derivative and anti derivative?
Antiderivatives are the opposite of derivatives. An antiderivative is a function that reverses what the derivative does. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals.
What are the basic formulas for integration involving trigonometric functions?
Integrals of Trigonometric Functions
| Function | Integral |
|---|---|
| cosx | sinx + c |
| sin2x | x/2 – sin(2x)/4 + c = (x – sinx ∙ cosx)/2 + c |
| cos2x | x/2 + sin(2x)/4 + c = (x + sinx ∙ cosx)/2 + c |
| tanx = sec2x | -ln|cosx| + c |
What are the differentiation formulas for finding the derivative of trigonometric functions?
Derivatives of Trigonometric Functions
| Function | Derivative |
|---|---|
| arccosx = cos-1x | -1/√(1-x2) |
| arctanx = tan-1x | 1/(1+x2) |
| arccotx = cot-1x | -1/(1+x2) |
| arcsecx = sec-1x | 1/(|x|∙√(x2-1)) |
What is anti-differentiation in calculus?
Anti-differentiation or integration is the reverse process to differentiation. For example, if f (x) = 2x, we know that this is the derivative of f(x) = x2. Note: Where possible, check your answer by differentiating, remembering that the derivative of a constant, c, is zero.
Why do we need antiderivative?
The area under the function (the integral) is given by the antiderivative! That is to say, if your function has a kink in it (the way |x| has a kink at zero, for example) then you can’t find a derivative at that kink, but integrals don’t have that problem.
What is integration of trigonometric functions?
Some integrals involving trigonometric functions can be evaluated by using the trigonometric identities. These allow the integrand to be written in an alternative form which may be more amenable to integration. use trigonometric identities to integrate sin2 x, cos2 x, and functions of the form sin 3x cos 4x.
What are the derivatives of a trig function?
The Quotient Rule Derivatives of Trig Functions Necessary Limits Derivatives of Sine and Cosine Derivatives of Tangent, Cotangent, Secant, and Cosecant Summary The Chain Rule
Are there any antiderivatives for tangent and cotangent?
The antiderivatives of tangent and cotangent are easy to compute, but not so much secant and cosecant. If playback doesn’t begin shortly, try restarting your device.
Can you use a trig formula to find the limit?
Since we can’t just plug in h = 0 h = 0 to evaluate the limit we will need to use the following trig formula on the first sine in the numerator. As you can see upon using the trig formula we can combine the first and third term and then factor a sine out of that.
Is there a similar proof for differentiating cosine?
Differentiating cosine is done in a similar fashion. It will require a different trig formula, but other than that is an almost identical proof. The details will be left to you.