integral of (1-tan^2x)/sec^2x, calculus 2 tutorial, trig integrals with trig identities

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Calculus 2, integral of (1-tan^2x)/sec^2x, integral of cos(2x)

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Wanted to share my method for this one: sec^2(x) = tan^2(x)+1. So you get tan^2(x)-1 / tan^2(x)+1, which is another identity for cos(2x) already. So, from the start, you can already transform the integral, with identities only, into Int((cos(2x)dx).

etiennedube

lol. Got the same result but used a different approach. converted tangent and secant into their sin/cos equivalents and then did a little algebra to get cos^2(x) - sin^2(x) then split it up and used power reduction formulas to simplify. Always fun when there are multiple paths to the same solution.

pipertripp

for those of you who are wondering how he got 2cos^2x-1 is because he split the 2-sec^2x and canceled sec^2x over sec^2x

MotivationalContent-rz

This is potentially good news for me. I am a bit like the ancient mariner in the famous poem - 'He stoppeth one in three'. But my problem is not to do with Albatross, it's about the integral of 1 + Tan^2 (x). So in 1972 when I did U.K. A level maths, we worked out the integral and I thought to myself, 'I wonder how that is going to impact my life?' I didn't exactly phrase it in the way I've written here, largely because back in those days I was a raw numpty. So I'm always pleased to see math videos on Youtube, in the hope that I can find out how the integral of 1 + Tan^2 (x) affects our daily lives. I never did find out and I'm 70 years old now....:)

Greebstreebling

When i see stuff like this i just think
-> convert into Trig rational form (sin and cos) -> weierstrass substitution -> Polynomial rational -> Partial Fraction

maalikserebryakov

THANK YOU SO MUCH! Your videos have been so helpful!

marthavilla

or write the numerator as (1+tan^2(x)-2tan^(x)/sec^2 (x)
1+tan^2 (x) is just sec^2(x)
so we are left with 1-(2tan^2x/sec^2x)
tan^2 (x)/sec^2(x) is just sin^2(x)
so only 1-2sin^2(x) is left
then integrate to get x-(x-sin2x/2)
which equals to sin2x/2 which further simplifies into sin(x)cos(x)

oh and dont forget the constant C ;)

Gawkie

I think it's much easier and faster to solve if you separate it into 2 fractions.

saeedsaleh

Al menos le entendí mas que a los profesores latinos

danielalopezuribe

I multiplied cos^2(x) top and bottom.

Then get cos^2(x) - sin^2(x) = cos(2x)

And then the same as you

joaquingutierrez

I DONT GET IT WDYM 2COS^2-1 IS HOW IM CRYING BRO.... IS IT THE HALF ANGLE IDENTITY OR WHAT IM SUPER I HATE CALCULUS

marya-d

integral of (1-tan^2x)/sec^2x, calculus 2 tutorial, trig integrals with trig identities

integral of (1-tan^2x)/sec^2x, calculus 2 tutorial, trig integrals with trig identities

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