Question

Вычислите неопределенный интеграл: x^2*x^1/2 dx; (8sin x - 9/cos^2 x)dx; cos8x * sin 6x dx; (4x-7)^6dx; v(t)=3t^2-4t+1 .

Answer 1

1. $\int{x^2\sqrt{x}}\, dx\ =\ \int{x^{\frac{5}{2}}}\, dx\ =\ \frac{2x^{\frac{7}{2}}}{7}.$

2. $\int{(8sinx-\frac{9}{cos^2x})}\, dx\ =\ -8cosx-9tgx.$

3. $\int{cos8x*sin6x}\, dx\ =\ \frac{1}{2}\int{sin14x}\, dx\ -\ \frac{1}{2}\int{sin2x}\, dx\ =$

= $-\frac{1}{28}cos14x\ +\ \frac{1}{4}cos2x\ =\ \frac{1}{28}(7cos2x\ -\ cos14x).$

4. $\int{(4x-7)^6}\, dx\ =\ \frac{1}{4}\int{(4x-7)^6}\, d(4x-7)\ =\ \frac{1}{28}(4x-7)^7.$

5. $\int{(3t^2-4t+1)}\, dt\ =\ t^3-2t^2+t.$