Question

Найти первообразную
f(x)=6/5√4x+2+1/cos^2 5x

Answer 1

Ответ:

$\frac85x\sqrt x + 2x + \frac15 \text{tg}\, 5x + C.$

Объяснение:

$f(x)=\frac65\sqrt{4x} + 2 + \frac{1}{\cos^2 5x};\\F(x) = \int \frac65\sqrt{4x} + 2 + \frac{1}{\cos^2 5x}\, \text{d}x;\\\\\int \frac65\sqrt{4x} + 2 + \frac{1}{\cos^2 5x}\, \text{d}x = \frac65 \int \sqrt{4x} \, \text{d}x + 2 \int \text{d} x + \int \frac{\text{d}x}{\cos^2 5x};\\\\ 1)\ \frac65 \int \sqrt{4x} \, \text{d}x = \frac652\int\sqrt{x}\, \text{d}x = \frac{12}5 \int x^\frac12 \, \text{d}x = \frac{12\cdot 2x^\frac32}{3\cdot5} = \frac85x^\frac32 = \frac85x\sqrt x + C.\\\\2)\ 2\int \text{d}x = 2x + C.$

$3)\ \int \frac{\text{d} x}{\cos^2 5x} = \frac{1}{5} \int \frac{\text{d}(5x)}{\cos^25x} = \frac15 \text{tg}\,5x + C.\\\\F(x) = \frac85x\sqrt x + 2x + \frac15 \text{tg}\, 5x + C.$