integral of (e^x+2sec^2(x)+4/x+sqrt(x))

解答

\int (e^{x} + 2 sec^{2} (x) + \frac{4}{x} + x) d x

e^{x} + 2 tan (x) + 4 ln ∣ x ∣ + \frac{2}{3} x^{\frac{3}{2}} + C

求解步骤

\int e^{x} + 2 sec^{2} (x) + \frac{4}{x} + x d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{x} d x + \int 2 sec^{2} (x) d x + \int \frac{4}{x} d x + \int x d x

\int e^{x} d x = e^{x}

\int 2 sec^{2} (x) d x = 2 tan (x)

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int x d x = \frac{2}{3} x^{\frac{3}{2}}

= e^{x} + 2 tan (x) + 4 ln ∣ x ∣ + \frac{2}{3} x^{\frac{3}{2}}

解答补常数

= e^{x} + 2 tan (x) + 4 ln ∣ x ∣ + \frac{2}{3} x^{\frac{3}{2}} + C

\int cot (x) (2 + ln ∣ sin (x) ∣) d x

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\int \frac{1}{x ^{2} + 10 x + 30} d x