integral of 2cos^2(x/2)

解答

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

x + sin (x) + C

求解步骤

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

使用三角恒等式改写

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int 1 + cos (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

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

\int 1 d x = x

\int cos (x) d x = sin (x)

= 2 \cdot \frac{1}{2} (x + sin (x))

化简

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

= x + sin (x)

解答补常数

= x + sin (x) + C

\int \frac{1}{t ln ( t )} d t

t an g e n t y = 5 e^{x} + x, (0, 5)

\int \frac{1}{49 - 64 x ^{2}} d x

\int t^{7} e^{- t^{4}} d t

\frac{d}{d x} (\frac{3 x ^{4} + 3}{sin ( x )})