Неопределенный интеграл. Непосредственное интегрирование. Метод замены переменных. Метод интегрирования по частям

1) INL {6sin(x) + 4x^3 - 1/x} * dx
=
6 * INL sin(x)*dx + 4 * INL x^3*dx - INL dx/x
=
6 * {-cos(x)} + 4 * x^4/4 - ln(x) + C
=
x^4 - 6cos(x) - ln(x) + C ,

2) INL {x^9 - 3x^7 + 2x^6} / x^7 * dx
=
INL {x^2 - 3 + 2/x} * dx
=
x^3/3 - 3x + 2ln(x) + C ,

3) INL {7^x * 2^(2x) + 5} * dx
=
INL {28^x + 5} * dx
=
INL e^{x*ln(28)} * dx + INL d(5x)
=
e^{x*ln(28)} : ln(28) + 5x + C
=
28^x : ln(28) + 5x + C ,

4) INL {1/(1+x^2) + 1/sin(x)^2} * dx
=
INL dx/(1+x^2) + INL dx/sin(x)^2
=
arctg(x) - ctg(x) + C ,

5) INL dx/sqrt(4-9x^2)
=
INL dx / 2sqrt{1-(3x/2)^2}
=
1/2 * INL 2/3*d(3x/2) / sqrt{1-(3x/2)^2}
=
1/3 * INL d(3x/2) / sqrt{1-(3x/2)^2}
=
1/3 * arcsin(3x/2) + C .
-------------------------------------------
1) INL (7x+5)^4 * dx
=
INL (7x+5)^4 * 1/7*d(7x+5)
=
1/7 * INL t^4 * dt
=
1/7 * t^5/5 + C
=
1/35 * (7x+5)^5 + C ,

2) INL (18x^2-3)/(6x^3-3x+8) * dx
=
INL d(6x^3-3x+8) / (6x^3-3x+8)
=
INL dt / t
=
ln(t) + C
=
ln(6x^3-3x+8) + C ,

3) INL {x^7 * e^(x^8)} * dx
=
INL e^(x^8) * 1/8*d(x^8)
=
1/8 * INL e^t * dt
=
1/8 * e^t + C
=
1/8 * e^(x^8) + C .
----------------------------

INL (x-2)sin(x)dx
=
-INL (x-2) * d{cos(x)}
=
-[(x-2)*cos(x) - INL cos(x) * d(x-2)]
=
INL cos(x) * dx - (x-2)*cos(x)
=
sin(x) - (x-2)*cos(x) + C .