Question

Интеграл 6x^2-8x^3+4

Answer 1

∫(6x^2-8x^3+4)dx=$\frac{ 6x^{3} }{3} - \frac{ 8x^{4} }{4} +4x=2x^{3}-2x^{4}+4x$

Answer 2

I can help you with math problems. To find the indefinite integral of 6x^2-8x^3+4, we can use the power rule of integration. The power rule states that $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$, where C is an arbitrary constant. Applying this rule to each term of the polynomial, we get:

$$\int 6x^2-8x^3+4 dx = 6 \int x^2 dx - 8 \int x^3 dx + 4 \int dx$$ $$= 6 \frac{x^3}{3} - 8 \frac{x^4}{4} + 4x + C$$ $$= 2x^3 - 2x^4 + 4x + C$$

This is the general antiderivative of the function. You can check your answer by taking the derivative of the result and verifying that it matches the original function.

If you want to find the definite integral of the function over a specific interval, you can use the fundamental theorem of calculus. The theorem states that $$\int_a^b f(x) dx = F(b) - F(a)$$, where F(x) is any antiderivative of f(x). For example, if you want to find the area under the curve from x = 0 to x = 1, you can plug in the values into the antiderivative and subtract:

$$\int_0^1 6x^2-8x^3+4 dx = (2x^3 - 2x^4 + 4x + C) \bigg|_0^1$$ $$= (2 - 2 + 4 + C) - (0 - 0 + 0 + C)$$ $$= 4$$

The area under the curve is 4 square units. Note that the constant C cancels out in the subtraction, so it does not affect the value of the definite integral.

If you need more help with integration, you can use one of these online calculators , , or . They can show you the steps and the graph of the function. I hope this was helpful. Have a nice day!