integral of x(sec^2(x)+2tan^2(x))

解答

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

- x^{2} + 3 x tan (x) + 3 ln ∣ cos (x) ∣ + C

求解步骤

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

使用三角恒等式改写

= \int \frac{x ( 1 + 2 sin ^{2} ( x ) )}{cos ^{2} ( x )} d x

乘开

\frac{x ( 1 + 2 sin ^{2} ( x ) )}{cos ^{2} ( x )} : \frac{x}{cos ^{2} ( x )} + \frac{2 x sin ^{2} ( x )}{cos ^{2} ( x )}

= \int \frac{x}{cos ^{2} ( x )} + \frac{2 x sin ^{2} ( x )}{cos ^{2} ( 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 \frac{x}{cos ^{2} ( x )} d x + \int \frac{2 x sin ^{2} ( x )}{cos ^{2} ( x )} d x

\int \frac{x}{cos ^{2} ( x )} d x = x tan (x) + ln ∣ cos (x) ∣

\int \frac{2 x sin ^{2} ( x )}{cos ^{2} ( x )} d x = - x^{2} + 2 x tan (x) + 2 ln ∣ cos (x) ∣

= x tan (x) + ln ∣ cos (x) ∣ - x^{2} + 2 x tan (x) + 2 ln ∣ cos (x) ∣

化简

= - x^{2} + 3 x tan (x) + 3 ln ∣ cos (x) ∣

解答补常数

= - x^{2} + 3 x tan (x) + 3 ln ∣ cos (x) ∣ + C

\int 35 x e^{- 7 x} d x

\frac{d}{d x} (cosh (\frac{e ^{x} + e ^{- x}}{2}))

\frac{d}{d x} (- 2 \cdot sin (2 x))

\int \frac{2 + 5 x}{1 + x ^{2}} d x

\frac{d}{d x} ((- 5 x)^{30})