Question

Найти F(х):а) f(x)=2x^2-3x+5б) f(x)=1/x-1/√xв) f(x)=cosx+2sinxг) f(x)= e^x+2^x

Answer 1

Решение смотрите во вкладке.

Answer 2

a) f(x) = 2x^2 - 3x + 5

To find the function F(x) for f(x) = 2x^2 - 3x + 5, we can use the power rule of integration. The power rule states that if we have a term of the form x^n in the integrand, then the integral of x^n with respect to x is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.

In this case, we have f(x) = 2x^2 - 3x + 5. To find F(x), we integrate each term separately:

∫(2x^2) dx = (2/3)x^3 + C1, where C1 is the constant of integration for the first term.

∫(-3x) dx = (-3/2)x^2 + C2, where C2 is the constant of integration for the second term.

∫(5) dx = 5x + C3, where C3 is the constant of integration for the third term.

Combining these results, we get:

F(x) = (2/3)x^3 - (3/2)x^2 + 5x + C, where C = C1 + C2 + C3 is the constant of integration for the entire function.

So, the function F(x) for f(x) = 2x^2 - 3x + 5 is F(x) = (2/3)x^3 - (3/2)x^2 + 5x + C.

b) f(x) = 1/x - 1/√x

To find the function F(x) for f(x) = 1/x - 1/√x, we can use the power rule of integration, as well as the rule for integrating the reciprocal of a function.

First, let's rewrite the function as f(x) = x^(-1) - x^(-1/2).

Integrating each term separately:

∫(x^(-1)) dx = ln|x| + C1, where C1 is the constant of integration for the first term.

∫(x^(-1/2)) dx = 2x^(1/2) + C2, where C2 is the constant of integration for the second term.

Combining these results, we get:

F(x) = ln|x| + 2x^(1/2) + C, where C = C1 + C2 is the constant of integration for the entire function.

So, the function F(x) for f(x) = 1/x - 1/√x is F(x) = ln|x| + 2x^(1/2) + C.

c) f(x) = cos(x) + 2sin(x)

To find the function F(x) for f(x) = cos(x) + 2sin(x), we can use the trigonometric identities and the rules for integrating trigonometric functions.

The integral of cos(x) is sin(x) + C1, where C1 is the constant of integration for the first term.

The integral of sin(x) is -cos(x) + C2, where C2 is the constant of integration for the second term.

Combining these results, we get:

F(x) = sin(x) - cos(x) + C, where C = C1 + C2 is the constant of integration for the entire function.

So, the function F(x) for f(x) = cos(x) + 2sin(x) is F(x) = sin(x) - cos(x) + C.

d) f(x) = e^x + 2^x

To find the function F(x) for f(x) = e^x + 2^x, we can use the rule for integrating exponential functions.

The integral of e^x is e^x + C1, where C1 is the constant of integration for the first term.

The integral of 2^x can be found using the substitution method or the rule for integrating exponential functions.

If we use the substitution method, we can let u = 2^x, then du = ln(2) * 2^x dx. Rearranging, dx = (1/ln(2)) * (1/u) du.

Substituting these values into the integral, we get:

∫(2^x) dx = ∫(u) * (1/ln(2)) * (1/u) du = (1/ln(2)) ∫(1) du = (1/ln(2)) * u + C2, where C2 is the constant of integration for the second term.

Substituting u = 2^x back in, we have:

∫(2^x) dx = (1/ln(2)) * 2^x + C2.

Combining these results, we get:

F(x) = e^x + (1/ln(2)) * 2^x + C, where C = C1 + C2 is the constant of integration for the entire function.

So, the function F(x) for f(x) = e^x + 2^x is F(x) = e^x + (1/ln(2)) * 2^x + C.