Can someone help me solve this anti derivative equation?

Question

I need to find the indefinite integral of
[Sin(x) / Cos^8 (x) ] dx

Answer 1

Take cos(x) = z then dz/dx = -sin(x) or, dz = -sin(x)dx
then the integration becomes [ -1/z^8 dz] = 1/(7z^7) = 1/(7cos^7(x))

Answer 2

Try using a substitution - are you familiar with this method?

Set u = cos(x)
hence, du = -sin(x)dx

Subbing this in to the original question gives:

Í(sin(x)/(cos^8(x)) dx = Í -1/(u^8) du

et. cetera - simply integrate that with respect to "u" and Bob's your uncle!

Hope that helped!

Answer 3

I = ∫ sin x / (cos x)^8 dx
Let u = (cos x)
du = (- sin x) dx
I = - ∫ 1 / u^8 du
I = - ∫ u^(- 8) du
I = (1/7) (1/u^7) + C
I = 1 / [ 7cos ^7 (x) ] + C

Answer 4

the respond is ln(cosx) - ln(cos(x) -a million) + C i think of I did it "the good way" and it took many substitutions. First, use the identity sin^2x + cos^2x = a million to simplify the indispensable. bear in mind: cosx = sqrt(a million-sin^2x) and cos^2x= a million- sin^2x sinx/ (a million-sin^2x) -cosx dx enable u= sinx and dx= du/ (cosx) [u/ ((a million- u^2) -cosx)] (du/ (cosx)) u/ ((a million-u^2)cosx - cos^2x) du u/ ((a million-u^2)cosx - (a million-u^2)) du u/ ((a million-u^2) (cosx -a million)) du u/ ((a million-u^2)(sqrt(a million-u^2) -a million)) du then enable v= a million- u^2 which makes -dv/(2u) = du u/((v) (sqrt(v)-a million)) (-dv/(2u)) which simplifies to -a million/2 [a million /((v) (sqrt(v)-a million)] now enable y= sqrt(v) -a million which makes dy(2sqrt(v))= dv -a million/2 [(a million /(vy)) (2sqrt(v))] dy which simplifies to -a million/2 [2/ (sqrt(v))y)] dy from above you comprehend that y + a million = sqrt(v) -a million/2 [2/ (y+a million)(y)] use partial fractions to get -a million/2 [-2 /(y+a million) + 2 /y] combine and multiply by means of -a million/2 to get ln (y +a million) - ln (y) now replace each and every thing lower back in... ln (sqrt(v)-a million +a million) - ln (sqrt(v) -a million) ln (sqrt(a million-u^2)) - ln (sqrt(a million-u^2) -a million) ln (sqrt(a million- sin^2x)) - ln (sqrt(a million- sin^2x) -a million)) by means of the identity sin^2x + cos^2x =a million, sqrt(a million- sin^2x) = cos x so the terrific answer is, ln(cosx) - ln(cos(x) -a million) + C

Answer 5

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