Question

Obliczyć całki...
10 całek do obliczenia.

potrzebuje to miec obliczone w kilku krokach,
a nie calka = wynik.

stad daje 40 punktow temu kto umie to rozwiazac.

http://w979.wrzuta.pl/obraz/7bhu0SzjnIS/calki

Answer 1

A)

∫(x² - 1)/x³ dx = ∫x²/x³ dx - ∫dx/x³ =

= ∫dx/x - ∫x⁻³dx =

= ln|x| + ½x⁻² + C =

= ln|x| + 1/2x² + C

b)

∫[(x² + 1)²]/x³ dx = ∫(x⁴ + 2x² + 1)/x³ dx =

= ∫x⁴/x³dx + 2∫x²/x³dx + ∫dx/x³ =

= ∫xdx + 2∫dx/x -1/2x² =

= 0,5x² + 2ln|x| - 1/2x² + C =

= 0,5x² + ln|x²| - 1/2x² + C

c)

∫(3 - 2x)⁴dx = { 3 - 2x = t , -2dx = dt , dx = - 0,5dt } = - 0,5*∫t⁴dt =

= - 0,5*1/5*t⁵ =

= - 0,1(3 - 2x)⁵ + C

d)

∫(1/∛x² - 1/x√x)dx = ∫(x^⁻2/3 - x ^⁻3/2)dx =

= ∫(x^⁻2/3)dx - ∫(x ^⁻3/2)dx =

= 3∛ +2/√x + C

e)

∫(e^½x - e^⁻½x)dx = ∫(e^½x)dx - ∫(e^⁻½x)dx =

= { e^½x = t , - ½( e^½x)dx = dt , ( e^½x)dx = 2dt }

{ e^⁻½x = w , ½( e^⁻½x)dx = dw , ( e^⁻½x)dx = - 2dw } =

= 2*∫tdt + 2*∫wdw =

= 2*0,5*t² + 2*0,5*w² =

= t² + w² =

= (e^½x)² + (e^⁻½x)² + C =

= e^x + 1/e^x + C

f)

∫(3 -2ctg²x)/cos²xdx = 3∫dx/cos²x - 2∫ctg²x/cos²xdx =

= 3tgx - 2∫[(cos²x/sin²x) : (cos²x)]dx =

= 3tgx - 2∫[(cos²x/sin²x) * (1/cos²x)]dx =

= 3tgx - 2∫dx/sin²x =

= 3tgx - 2( - ctgx) + C =

= 3tgx + 2ctgx + C

g)

∫ctg²xdx = - ctgx - ∫ctg⁺xdx =

= - ctgx - ∫dx =

= - ctgx - x + C