Hvordan integrerer du int x + cosx fra [pi / 3, pi / 2]?

Svar:

Svaret #int _ (pi / 3) ^ (pi / 2) x + cosx * dx = 0,8193637907356557 #

Forklaring:

Vis nedenfor

(pi / 3) ^ (pi / 2) x + cosx * dx = 1 / 2x ^ 2 + sinx _ (pi / 3) ^ (pi / 2)

# Pi ^ 2 / på 8 + sin (pi / 2) - pi ^ 2 / 18+ sin (pi / 3) = (5 * pi ^ 2-4 * 3 ^ (5/2) +72) /72=0.8193637907356557#

Svar:

(pi / 2) (x + cosx) dx = 1 + (5pi ^ 2-36sqrt3) / 72 #

Forklaring:

Brug af integralens linearitet:

pi / 3) ^ (pi / 2) (x + cosx) dx = int_ (pi / 3) ^ (pi / 2) xdx + int_ (pi / 3) ^ (pi / 2) cosxdx #

Nu:

(pi / 3) ^ (pi / 2) xdx = x ^ 2/2 _ (pi / 3) ^ (pi / 2) = pi ^ 2/8-pi ^ 2/18 = 2) / 72 #

(pi / 2) = sin (pi / 2) -sin (pi / 3) = 1-kvm3 / 2 (pi / 2) #

Derefter:

(pi / 2) (x + cosx) dx = 1 + (5pi ^ 2-36sqrt3) / 72 #

Svar:

# (5π ^ 2) / 72 + 1-sqrt3 / 2 #

Forklaring:

#int_ (π / 3) ^ (π / 2) (x + cosx) dx # #=#

#int_ (π / 3) ^ (π / 2) xdx + int_ (π / 3) ^ (π / 2) cosxdx # #=#

# X ^ 2/2 _ (π / 3) ^ (π / 2) # #+# # Sinx _ (pi / 3) ^ (π / 2) # #=#

# (Π ^ 2/4) / 2- (π ^ 2/9) / 2 + sin (π / 2) -sin (π / 3) # #=#

# π ^ 2/8-π ^ 2/18 + 1-sqrt3 / 2 # #=#

# (5π ^ 2) / 72 + 1-sqrt3 / 2 #

Hvordan bevise (1 + sinx-cosx) / (1 + cosx + sinx) = tan (x / 2)?

Se nedenfor. LHS = (1-cosx + sinx) / (1 + cosx + sinx) = (2sin ^ 2 (x / 2) + 2sin (x / 2) * cos (x / 2)) / (2cos ^ 2 2) + 2sin (x / 2) * cos (x / 2) = (2sin (x / 2) [sin (x / 2) + cos (x / 2)]) synd (x / 2) + cos (x / 2)]) = tan (x / 2) = RHS

Hvordan ville jeg gå om at bevise, at dette er en identitet? Tak skal du have. (1-sin ^ 2 (x / 2)) / (1 + sin ^ 2 (x / 2)) = (1 + cosx) / (3-cosx)

LHS = (1-sin ^ 2 (x / 2)) / (1 + sin ^ 2 (x / 2) = (cos ^ 2 (x / 2)) / (1 + 1-cos ^ 2 )) = (2cos ^ 2 (x / 2)) / (2-2cos2 2 (x / 2)) = (1 + cosx) / (4- (1 + cosx)) = (1 + cosx) / 3-cosx) = RHS

Bevis det: sqrt ((1-cosx) / (1 + cosx)) + sqrt ((1 + cosx) / (1-cosx)) = 2 / abs (sinx)?

Bevis under anvendelse af konjugater og trigonometrisk version af Pythagorean Theorem. Del 1 sqrt (1-cosx) / (1 + cosx)) farve (hvid) ("XXX") = sqrt (1-cosx) / sqrt (1 + cosx) farve (hvid) ("XXX") = sqrt (1-cosx)) / sqrt (1 + cosx) * sqrt (1-cosx) / sqrt (1-cosx) farve (hvid) ("XXX") = (1-cosx) / sqrt 2x) Del 2 Tilsvarende sqrt ((1 + cosx) / (1-cosx) farve (hvid) ("XXX") = (1 + cosx) / sqrt (1-cos ^ 2x) Del 3: Kombination af udtrykkene sqrt (1-cosx) / (1 + cosx)) + sqrt (1 + cosx) / (1-cosx) farve (hvid) ("XXX") = (1-cosx) / sqrt (1-cos ^ 2x) + (1 + cosx) / sqrt (1-cos ^ 2x