Question

Решите пожалуйста, буду очень благодарен!

Answer 1

1)

$int {3(x^3 - 1)^4x^2} , dx, t = x^3 - 1, dt = 3x^2dx\[tex]int {3(x^3 - 1)^4x^2} , dx=int {t^4}, dt = frac{t^5}{5} + C = frac{(x^3 - 1)^5}{5} + C$

2)

$int {frac{(1 - ln(x))^4}{x}} , dx, t = 1 - ln(x), dt = frac{dx}{x}\int {frac{(1 - ln(x))^4}{x}} , dx = int {t^4}, dt = frac{t^5}{5} + C = frac{(1 - ln (x))^5}{5} + C$

3)

$int {(5x + 4)^3} , dx, t = 5x + 4, dt = 5dx\int {(5x + 4)^3} , dx = frac{1}{5}int {t^3} , dt = frac{t^4}{20} + C = frac{(5x + 4)^4}{20} + C$

4)

$int {frac{x}{x^2 + 1}} , dx, t = x^2 + 1, dt = 2xdx\int {frac{x}{x^2 + 1}} , dx = frac{1}{2} int {frac{1}{t}} , dt = frac{ln(t)}{2} + C = frac{ln(x^2 + 1)}{2} + C$

5)

$int {e^{cos(x)}sin(x)} , dx, t = cos(x), dt = -sin(x)dx\int {e^{cos(x)}sin(x)} , dx = -int {e^t} dt = -e^t + C$