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Задать вопрос экспертам

Консультация № 174365

19.11.2009, 12:03

35.00 руб.

0 2 2

Образование

Здравствуйте, Уважаемые эксперты.
Нужно найти интегралы

1. [$8747$]cos(3-5x) dx
2. [$8747$](x²*[$8730$](1+x³)) dx
3. [$8747$]([$8730$](x)*lnx) dx
4. [$8747$]dx/(x*(x-1)²)
5. [$8747$][$8730$](x+1)/(³[$8730$](x+1)+1) dx
6. [$8747$]dx/(1+sinx)

Буду очень благодарна, если решите хотя бы часть

Обсуждение

Неизвестный

19.11.2009, 14:53

общий

это ответ

Здравствуйте, nani120.
1. ∫cos(3-5x) dx=-1/5∫cos(3-5x)d(3-5x)=-1/5 sin(3-5x)+C
2. ∫x^2*√(1+x^3) dx=1/3∫√(1+x^3)d(1+x^3)=2/9√((1+x^3))^3 +C

давно

Академик
320937
2216

19.11.2009, 15:06

общий

это ответ

Здравствуйте, nani120.

3. Интегрирование «по частям». ∫udv = uv-∫vdu
∫(√(x)*lnx)dx = |√(x)dx=dv, v=∫√(x)dx=∫x^(1/2)dx=(2/3)x^(3/2), u=lnx, du=1/x|
= lnx*(2/3)*x^(3/2) – ∫(2/3)*x^(3/2)/xdx = (2/3)*x^(3/2)*lnx – (4/9)*x^(3/2)+С

4. ∫dx/(x*(x-1)²)
Разложим дробь на простейшие
1/(x*(x-1)²)=A/x+B/(x-1)+C/(x-1)²=(A(x-1)²+Bx(x-1)+Cx)/(x(x-1)²)
=(x²(A+B)+x(-2A-B+C)+A)/ (x(x-1)²)
A+B=0; -2A-B+C=0; A=1 =>
A=1, B=-1, C=1

∫dx/(x*(x-1)²)= ∫(1/x-1/(x-1)+1/(x-1)²)dx=ln|x|-ln|x-1|-1/(x-1)+C
= ln|x/(x-1)|-1/(x-1)+C

5. ∫√(x+1)/(3√(x+1)+1)dx=|(x+1)^(1/6)=t, x+1=t⁶, x=t⁶-1,dx=6t⁵dt|
=∫t³/(t²+1)* 6t⁵dt=6∫t⁸/(t²+1) dt=6∫(t⁸+ t⁶- t⁶- t⁴+ t⁴+ t²- t²-1+1)/(t²+1) dt
=6∫t⁶ - t⁴ + t² -1+1dt+6∫1/(t²+1) dt=6t⁷/7-6t⁵/5+6t³/3-6t+6arctg(t)+C=
=(6/7)(x+1)^(7/6)-6/5(x+1)^(5/6)+2(x+1)^(1/2)-6(x+1)^(1/6)+6arctg((x+1)^(1/6))+C=

6.∫dx/(1+sinx) = |tg(x/2)=t, x=arctg(2t), dx=dt/(1+t2), sinx=2sin(x/2)cos(x/2)=2cos2(x/2)*tg(x/2)=2tg(x/2)/(1+tg²(x/2))=2t/(1+t²)|
=∫dt/((1+t²)*(1+2t/(1+t²))= ∫dt/(1+t) ²=∫(1+t)^-2d(1+t)=-(1+t)-1+C=-1/(1+tg(x/2))+C

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