Question

1) ∫ dx/корень5x+2 2)∫ (5x^4+14x^2+9)/(x^2+2)dx 3)∫(3корня из x - 1/x^3 - x^8)dx 4)∫(3-2x)/(2+x)dx 5)∫(1+3x)/(корень 1-4x^2+2x)dx 6)∫ (arctg2x)/(1+4x^2)dx 7)∫(x^2dx)/(sin^2 5x^3)

Answer 1

$\displaystyle\int \frac{dx}{\sqrt{5x+2}}=\frac{1}{5}\int\frac{d(5x+2)}{\sqrt{5x+2}}=\frac{2}{5}\sqrt{5x+2}+C$

$\displaystyle\int \frac{5x^4+14x^2+9}{x^2+2}dx=\int(5x^2+4+\frac{1}{x^2+2})dx=\frac{5x^3}{3}+4x+\\+\frac{1}{\sqrt2}arctg\frac{x}{\sqrt2}+C$

$\displaystyle \int(3\sqrt{x}-\frac{1}{x^3}-x^8)dx=2\sqrt{x^3}+\frac{1}{2x^2}-\frac{x^9}{9}+C$

$\displaystyle \int\frac{3-2x}{2+x}dx=-2\int\frac{-\frac{7}{2}+x+2}{2+x}=7\int\frac{d(x+2)}{x+2}-2\int dx=\\=7ln|x+2|-2x+C$

$\displaystyle \int \frac{1+3x}{\sqrt{1-4x^2+2x}}dx=-\frac{3}{8}\int\frac{-\frac{14}{3}-8x+2}{\sqrt{1-4x^2+2x}}=\\=-\frac{3}{8}\int\frac{d(1-4x^2+2x)}{\sqrt{1-4x^2+2x}}+\frac{7}{8}\int\frac{d(2x-\frac{1}{2})}{\sqrt{-(2x-\frac{1}{2})^2+\frac{5}{4}}}=\\=-\frac{3}{4}\sqrt{1-4x^2+2x}+\frac{7}{8}arcsin\frac{\sqrt{4}(4x-1)}{2\sqrt{5}}+C\\(1-4x^2+2x)'=-8x+2$

$\displaystyle \int\frac{arctg2x}{1+4x^2}dx=\frac{1}{2}\int arctg2xd(arctg2x)=\frac{arctg^22x}{4}+C$

$\displaystyle \int \frac{x^2dx}{sin^25x^3}=\frac{1}{15}\int\frac{d(5x^3)}{sin^25x^3}=-\frac{1}{15}ctg5x^3+C$

Answer 2

1. To solve ∫ dx/корень5x+2, we can use the substitution u = 5x + 2, which gives us du/dx = 5 and dx = du/5. Substituting these values into the integral, we get:

∫ dx/корень5x+2 = ∫ (1/кореньu)(du/5) = (1/5)∫ u^(-1/2)du = (1/5) * 2u^(1/2) + C = (2/5)*корень(5x+2) + C, where C is the constant of integration.

2. To solve ∫ (5x^4+14x^2+9)/(x^2+2)dx, we can use polynomial division or partial fractions to simplify the integrand. Here, we will use polynomial division:

   markdownCopy code
       5x^2 - 6
   

   ---

x^2 + 2 | 5x^4 + 14x^2 + 9 - 5x^2 - 10 _____________ 4x^2 + 9

Now, we can rewrite the integral as:

∫ (5x^4+14x^2+9)/(x^2+2)dx = ∫ (5x^2 - 6)dx + ∫ (4x^2 + 9)/(x^2+2)dx

The first integral is easy to solve:

∫ (5x^2 - 6)dx = (5/3)x^3 - 6x + C1, where C1 is the constant of integration.

To solve the second integral, we can use the substitution u = x^2 + 2, which gives us du/dx = 2x and dx = du/(2x). Substituting these values into the integral, we get:

∫ (4x^2 + 9)/(x^2+2)dx = ∫ (4u-14)/(2u) * (du/(2x)) = 2∫ (2u-7)/(ux)du = 2ln|ux| - 7(2/5) * (1/u^(5/2)) + C2 = 2ln|x^2+2| - (14/5) * (1/(x^2+2)^(5/2)) + C2, where C2 is the constant of integration.

Therefore, the complete solution is:

∫ (5x^4+14x^2+9)/(x^2+2)dx = (5/3)x^3 - 6x + 2ln|x^2+2| - (14/5) * (1/(x^2+2)^(5/2)) + C, where C is the constant of integration.

3. To solve ∫(3корня из x - 1/x^3 - x^8)dx, we can split the integral into three parts and solve them separately.

∫(3корня из x)dx = (2/5)x^(5/2) + C1, where C1 is the constant of integration.

∫(1/x^3)dx = (-1/2x^2) + C2, where C2 is the constant of integration.

To solve ∫ x^(-8) dx, we can use the substitution u = x^(-7), which gives us du/dx = -7x^(-8) and