We are simply referring to one of many possible functions whose derivatives equal f For example, x3 is an antiderivative of 3x2 f (g (x)) = tan (2x) ⇒ f' (g (x)) = sec2(2x) = 2sec 2 (2x) Using the chain rule, the derivative of tan (2x) is 2sec2(2x) Finally, just a note on syntax and notation tan (2x) is sometimes written in the forms below (with the derivative as per the calculation above) Just be aware that not all of the forms below are mathematically correct tan2x The integral of tan x, also known as the antiderivative of tan x, is a result that many calculus students and mathematicians memorizeUnfortunately, sometimes you forget it and need to derive it Or maybe, just maybe, you want to prove it to yourself Before we look at deriving the integral of tan x, let's first look at the end result, to see where we're heading
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Tan^2x antiderivative
Tan^2x antiderivative-C being a constant So integration of tan^2 (x) would be tanx x c;To integrate tan^22x, also written as ∫tan 2 2x dx, tan squared 2x, (tan2x)^2, and tan^2(2x), we start by utilising standard trig identities to change the form of the integral Our goal is to have sec 2 2x in the new form because there is a standard integration solution for that in formula booklets that we can use We recall the Pythagorean trig identity, and multiply the angles by 2
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O tan (x2 x) 1 tan (22 x 1) 01 tan (2x 1) 3 tan (2 1) 1 tan (2c 1) 1 Question Which one of the following is an antiderivative of the function f (x)=sec (2x1) ? Example 14 Show that tan 3𝑥 tan 2𝑥 tan 𝑥 = tan 3𝑥 – tan 2𝑥 – tan 𝑥 We know that 3𝑥 = 2𝑥 𝑥 Therefe, tan 3𝑥 = tan(2𝑥 𝑥Differentiate c and d, use the product rule to find v Then just use the product rule on u and v 0
Antiderivative of 2tan x sec x Compute tan x sec 2 x dx in two different ways a) By substituting u = tan x b) By substituting v = sec x c) Compare the two results Solution a) Compute tan x sec 2 x dx by substituting u = tan x If u = tan x 2then du = sec x dx and tan x sec 2 x dx = u du = 1 u2 c 2 = 1 tan2 x c 2 Your original integral is $$ \int\tan^2x\sec^2x\dx=\tfrac13\tan^3xC $$ Share Cite Follow edited Aug 26 ' at 617 answered Aug ' at 1445 JG JG 106k 6 6 gold badges 67 67 silver badges 126 126 bronze badges $\endgroup$ Add a commentKeep breaking it down until you find something you can work with Let u=sec^2 and v=tan^2 and if that's still too much at this stage Let a=sec b=sec c=tan d=tan Differentiate a and b, use the product rule to find u;
The Second Derivative Of tan^2x To calculate the second derivative of a function, differentiate the first derivative From above, we found that the first derivative of tan^2x = 2tan (x)sec 2 (x) So to find the second derivative of tan^2x, we need to differentiate 2tan (x)sec 2 (x)The solution is quite simple the antiderivative of 1/x is ln(x)The integral formula of tan is Of so many trigonometric integrals, we will see some examples for tangent integrals Example 1 Integral of tan 2x We substitute the 2 x for u, we derive and we pass dividing the 2 And we replace the terms for u By properties of integrals we extract the 1 2 from the integral Now we apply directly the tangent
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Solve the integral = ln u C substitute back u=cos x = ln cos x C QED 2 Alternate Form of Result tan x dx = ln cos x C = ln (cos x)1 C = ln sec x C ThereforeIntegral of tan^2 (x) \square!This would normally be quite a difficult integral to solve However, the ever powerful trigonometric identities make this an easy problem By realising that tan^2(x)=sec^2(x) 1, the integration of these two terms evaluates to tan(x) x Thanks for watching
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Click here👆to get an answer to your question ️ intsin4xe^tan^2x dxSo tan^2 (x)=sec^2 (x)1;integrate this equation on both sides w r to x We know that integration of sec^2 (x) is tanx c;You probably know the series for \tan^ {1} t Plug in 2x^2 for t If you do not know the series for \arctan t, you undoubtedly know the series for \dfrac {1} {1u} Set u=x^2, and integrate You probably know the series for tan−1t Plug in 2x2 for t If you do not know the series for arctant, you undoubtedly know the series for 1−u1
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Integrate the following {eq}f(x) = \tan(2x) {/eq} Integrals Now in this problem, we shall only apply the tangent standard antiderivative formula and the job is done Calculus questions and answers Which one of the following is an antiderivative of the function f (x)=sec (2x1) ?Learn how to solve trigonometric integrals problems step by step online Solve the trigonometric integral int(sec(2x)tan(2x))dx We can solve the integral \int\sec\left(2x\right)\tan\left(2x\right)dx by applying integration by substitution method (also called USubstitution) First, we must identify a section within the integral with a new variable (let's call it u), which when substituted
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