What is the anti derivative of (sinx) divided by (cos^2x)dx?

Question

Calculus Integrals

Answer 1

To solve this integral, you have to use substitution.

Integral ( sin(x) / cos^2(x) ) dx

I'm going to change the form of this question to make something obvious when I do the substitution.

Integral ([1/cos^2(x)]) [sin(x)] dx

Now, we begin our substitution.
Let u = cos(x). Then
du = -sin(x) dx, and
-du = sin(x) dx

Note the sin(x)dx attached at the end of our integral; I grouped them together specifically to show how (-du) would replace it entirely, due to what we just solved for. Now, our integral becomes

Integral (1/[u]^2)(-du)

Let's pull out that minus sign, which means we're effectively pulling out -1. We can *always* pull out constants in integrals.

(-1) * Integral (1/u^2) du

Remember that 1/u^2 is the same as u^(-2). Therefore, our integral becomes:

(-1) * Integral (u^(-2)) du

and now we use our reverse power rule.

(-1) [u^(-1)]/(-1) + C

reducing it further,

-1/u + C

And then substitution u = cos(x) back, we get

-1/cos(x) + C

Answer 2

1/cos(x)
this is an integral
so you can solve this integral by using a simpole substution
You let u = cos x
du = -sin x dx
your integral then becomes -1/u^2 = 1/u
change u and you get 1/cos(x)

Answer 3

it fairly is a confusing one, and is superb left memorized. vital ( a million/sin(x) dx ) that's comparable to vital ( csc(x) dx ) to unravel this, we predict of of this as a fragment over a million. vital ( csc(x)/a million dx ) Multiply precise and backside via csc(x) - cot(x). vital ( csc(x) [ csc(x) - cot(x) ] / [csc(x) - cot(x)] dx ) strengthen the numerator. vital ( [ csc^2(x) - csc(x)cot(x) ] / [ csc(x) - cot(x)] dx ) to coach how the substitution will artwork, i will rearrange this expression. vital ( a million / [ csc(x) - cot(x)] [ csc^2(x) - csc(x)cot(x) ] dx ) that's the place we use substitution. permit u = csc(x) - cot(x). Then du = [ -csc(x)cot(x) - (-csc^2(x)) ] dx du = [ -csc(x)cot(x) + csc^2(x) ] dx du = [ csc^2(x) - csc(x)cot(x) ] dx using our substitution to that end, this will become vital ( (a million/u) du ) Which we are able to unravel, as ln|u| + C back-substituting u = csc(x) - cot(x), we get ln|csc(x) - cot(x)| + C

Answer 4

(sinx / (cos^2x)) = sinx /[(cosx)(cosx)=(tanx)(sec x)
hence Antiderivative(sinx / (cos^2x))

Antiderivative (tanx)(sec x)
= Secx + c