Question

написать полное решение. найти неопределенный интеграл 1) S(sinx + 2/x) * dx 2) Se sin2xdx Найти определенные интегралы 1) S1^4(x^3 + x^2 + x/x^2) * dx 2) S0^пи/2 dx/1 - sin^2 x Найти S фигуры ограниченной линиямиy= корень из x; x1=1, y=0

Answer 1

1.=2logx-cosx+C

2.=1/2 e*x^2*sin(2)+C

1.=1/12*x^3(3x+4)+lnx=339/4+ln4

2.=tgx=0

Answer 2

Finding the Indefinite Integrals

1) To find the indefinite integral of ∫(sin(x) + 2/x) dx, we can split it into two separate integrals:

- ∫sin(x) dx - ∫2/x dx

The integral of sin(x) is -cos(x) + C, where C is the constant of integration. This can be verified using the derivative of -cos(x), which is sin(x).

The integral of 2/x can be found using the natural logarithm function. The integral is 2ln|x| + C.

Therefore, the indefinite integral of ∫(sin(x) + 2/x) dx is -cos(x) + 2ln|x| + C.

2) To find the indefinite integral of ∫e*sin(2x) dx, we can use integration by parts. Let's assume u = e and dv = sin(2x) dx. Then, we can find du and v using differentiation and integration, respectively.

- Differentiating u = e gives du = e dx. - Integrating dv = sin(2x) dx gives v = -1/2 cos(2x).

Now, we can apply the integration by parts formula:

∫u dv = uv - ∫v du

Plugging in the values, we get:

∫e*sin(2x) dx = -1/2 e*cos(2x) - ∫(-1/2 cos(2x)) e dx

Simplifying further, we have:

∫e*sin(2x) dx = -1/2 e*cos(2x) + 1/2 ∫cos(2x) e dx

The integral of cos(2x) e dx can be found using integration by parts again. Assuming u = cos(2x) and dv = e dx, we can find du and v using differentiation and integration, respectively.

- Differentiating u = cos(2x) gives du = -2 sin(2x) dx. - Integrating dv = e dx gives v = e.

Applying the integration by parts formula again, we get:

∫cos(2x) e dx = cos(2x) e - ∫(-2 sin(2x)) e dx

Simplifying further, we have:

∫cos(2x) e dx = cos(2x) e + 2 ∫sin(2x) e dx

We can substitute this result back into the previous equation:

∫e*sin(2x) dx = -1/2 e*cos(2x) + 1/2 (cos(2x) e + 2 ∫sin(2x) e dx)

Simplifying further, we have:

∫e*sin(2x) dx = -1/2 e*cos(2x) + 1/2 cos(2x) e