Question

Вычислить неопределенный интеграл С подробным решением

Answer 1

Упростим подынтегральное выражение, используя свойства степеней $x^n \cdot x^m=x^{n+m}, \ \ \ \sqrt[n]{x^m} = (\sqrt[n]{x})^m=x^{\frac{m}{n}}$

А неопределенный интеграл найдем по формуле $\int {x^n} \, dx= \frac{x^{n+1}}{n+1}+C$

$\int {x \cdot \sqrt{x}} \, dx = \int {x \cdot x^{\frac{1}{2}}} \, dx = \int {x^{\frac{3}{2}}} \, dx = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + C =\frac{2}{5} \cdot \sqrt{x^5}+C$

Answer 2

Calculating an indefinite integral with detailed solution

To calculate an indefinite integral, we must find an antiderivative (or primitive function) of the integrand. Let's consider the integral:

`∫(10x) dx`

The first step is to recognize that this is an elementary integral where the power rule can be used. Using the power rule, we know that the antiderivative of 10x is 10x^2/2. Therefore, the solution is:

`10x^2/2 + C`

Where C is the constant of integration.

Now let's consider a more complex example:

`∫(x^2 + 5x - 3) dx`

For integrals with polynomials, we can use the power rule repeatedly to find the antiderivative of each term separately and then add them together.

For the x^2 term: `x^3/3`

For the 5x term: `5x^2/2`

For the -3 term: `-3x`

Adding these together gives the full antiderivative as:

`x^3/3 + 5x^2/2 - 3x + C`

So in summary, to find an indefinite integral:

1. Identify the type of integral (polynomial, trigonometric, exponential etc.) 2. Use integration rules and properties to find the antiderivative term-by-term 3. Add the antiderivatives together 4. Include the constant of integration

I hope this detailed explanation of solving indefinite integrals was helpful! Let me know if you have any other questions.