integral of sin^2(pi/4+t/2)

解答

\int sin^{2} (\frac{π}{4} + \frac{t}{2}) d t

\frac{1}{2} (t - cos (t)) + C

求解步骤

\int sin^{2} (\frac{π}{4} + \frac{t}{2}) d t

使用三角恒等式改写

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

提出常数:

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

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

使用三角恒等式改写

= \frac{1}{2} \cdot \int 1 + sin (t) d t

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

= \frac{1}{2} (\int 1 d t + \int sin (t) d t)

\int 1 d t = t

\int sin (t) d t = - cos (t)

= \frac{1}{2} (t - cos (t))

解答补常数

= \frac{1}{2} (t - cos (t)) + C

x \to - 7 lim (x^{2} - 49)

d er i v a t i v e \frac{1}{t ^{2} + 5 t - 1}

t an g e n t y = \frac{4 x}{x ^{2} + 1}, (- 1, - 2)

a re a x^{3} - 13 x^{2} + 30 x, - x^{3} + 13 x^{2} - 30 x

l a pl a ce t r an s f or m - \frac{e ^{- 3 (t - 1)}}{3} - \frac{1}{3}