If ##x = 3\tan(t)##, then ##dx = 3\sec^2(t)dt = 3(\tan^2(t) + 1)dt##Memo said:
That was just from the step of taking the derivative of sin/cos.Mark44 said:
I guess aka ##sec^2 tdt##FactChecker said:
Why not just go directly to the derivative of the tangent function?FactChecker said:
I don't know. Your guess is as good as mine. Maybe they wanted to keep the required background to just derivative of sin, cos, and the quotient rule.Mark44 said:
The derivative of ##tan(x)## is ##\frac1{cos^2(x)}=sec^2(x)##.Mark44 said:
Yes, I realize that. My point was in explaining why ##x = 3\tan(t)## implies that ##dx = 3\sec^2(t)dt = 3(1 + \tan^2(t))dt##.martinbn said:
What was problem with the explanation ##dx = \frac{3dt}{cos^2(t)} = 3(1 + \tan^2(t))dt##.Mark44 said:
Assuming someone knows the derivatives of all six circular trig functions, why would someone write ##\frac{d\tan(t)}{dt}## as ##\frac 1 {\cos^2(t)}## and not just go directly to ##\sec^2(t)##?martinbn said:
Where I come from ##sec## and ##csc## are not used at all. For someone like me the derivative of tangent is one over cosine squared.Mark44 said:
I have taught college-level mathematics for 20+ years. Every calculus textbook I've ever seen presents all six circular trig functions as well as their derivatives.martinbn said:
Yes, and when I went to the US and thought calculus, I learned the notations for ##\frac1{\cos(x)}## and ##\frac1{\sin(x)}##. But where I grew up there were no such functions.Mark44 said:
Trig substitution is a technique used to simplify integrals by substituting trigonometric functions for certain algebraic expressions. It is particularly useful when dealing with integrals involving square roots of quadratic expressions. By using trigonometric identities, the integral can often be transformed into a simpler form that is easier to evaluate.
The choice of trigonometric substitution depends on the form of the expression under the square root:- For expressions of the form √(a^2 - x^2), use x = a sin(θ).- For expressions of the form √(a^2 + x^2), use x = a tan(θ).- For expressions of the form √(x^2 - a^2), use x = a sec(θ).
The steps involved are:1. Identify the appropriate trigonometric substitution based on the form of the integrand.2. Substitute the trigonometric function for the variable and simplify the integral.3. Integrate with respect to the new variable.4. Use trigonometric identities to simplify the result.5. Substitute back the original variable using the inverse trigonometric function.
When you perform a trigonometric substitution, you must also change the differential. For example, if x = a sin(θ), then dx = a cos(θ) dθ. This new differential must be included in the integral to correctly transform it into the new variable.
Sure! Let's solve the integral ∫ √(1 - x^2) dx.1. Use the substitution x = sin(θ), so dx = cos(θ) dθ.2. The integral becomes ∫ √(1 - sin^2(θ)) cos(θ) dθ.3. Simplify using the identity 1 - sin^2(θ) = cos^2(θ), so the integral becomes ∫ cos^2(θ) dθ.4. Use the identity cos^2(θ) = (1 + cos(2θ))/2 to further simplify: ∫ (1 + cos(2θ))/2 dθ.5. Integrate to get (θ/2) + (sin(2θ)/4) + C.6. Substitute back using θ = arcsin(x) to get (arcsin(x)/2) + (x√(1 - x^2)/2) + C.