Question

∫(1+√x)(1+кубический √x)dx

Answer 1

Ответ:

Пошаговое объяснение:

Answer 2

To evaluate the integral ∫(1+√x)(1+∛√x) dx, we can simplify the expression inside the integral first. Let's do that step by step:

1. Expand the expression: (1 + √x)(1 + ∛√x) = 1 + √x + ∛√x + √x * ∛√x

2. Now, let's substitute √x with a variable u to simplify further: Let u = √x Then, u^3 = (√x)^3 = x

Now, the integral becomes: ∫(1 + u)(1 + u^3) du

3. Expand and integrate: ∫(1 + u + u^3 + u^4) du = u + u^2/2 + u^4/4 + u^5/5 + C

4. Substitute back u with √x: ∫(1 + √x + (√x)^4/4 + (√x)^5/5) dx = ∫(1 + √x + x^2/4 + x^2√x/5) dx

Now, you have the indefinite integral in terms of x. If you want to find the definite integral over a specific interval, you can use these antiderivatives and apply the limits of integration.

Answer 3

To evaluate the integral ∫(1+√x)(1+∛√x)dx, we can use the technique of substitution. Let's set u = √x, then du = (1/2√x) dx. Also, the expression (1+∛√x) can be written as (1+u^(1/3)). Now, we can rewrite the integral in terms of u:

∫(1+√x)(1+∛√x) dx = ∫(1+u)(1+u^(1/3)) * 2u du (substituting u = √x and dx = 2u du)

Now, expand the expression:

∫(1+u)(1+u^(1/3)) * 2u du = ∫(2u + 2u^(4/3) + 2u^(1/3) + 2u^(2/3)) du

Integrate term by term:

∫(2u + 2u^(4/3) + 2u^(1/3) + 2u^(2/3)) du = u^2 + (3/5)u^(7/3) + (3/2)u^(4/3) + (3/2)u^(5/3) + C

Finally, don't forget to revert the substitution u = √x:

∫(1+√x)(1+∛√x) dx = √x^2 + (3/5)√x^(7/3) + (3/2)√x^(4/3) + (3/2)√x^(5/3) + C

So, the result of the integral is:

∫(1+√x)(1+∛√x) dx = x + (3/5)x^(7/6) + (3/2)x^(2/3) + (3/2)x^(5/6) + C

where C is the constant of integration.