integral of 3t^3(t^2+4)^5

解答

\int 3 t^{3} (t^{2} + 4)^{5} d t

\frac{3 t ^{14}}{14} + 5 t^{12} + 48 t^{10} + 240 t^{8} + 640 t^{6} + 768 t^{4} + C

求解步骤

\int 3 t^{3} (t^{2} + 4)^{5} d t

提出常数:

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

= 3 \cdot \int t^{3} (t^{2} + 4)^{5} d t

乘开

t^{3} (t^{2} + 4)^{5} : t^{13} + 20 t^{11} + 160 t^{9} + 640 t^{7} + 1280 t^{5} + 1024 t^{3}

= 3 \cdot \int t^{13} + 20 t^{11} + 160 t^{9} + 640 t^{7} + 1280 t^{5} + 1024 t^{3} 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

= 3 (\int t^{13} d t + \int 20 t^{11} d t + \int 160 t^{9} d t + \int 640 t^{7} d t + \int 1280 t^{5} d t + \int 1024 t^{3} d t)

\int t^{13} d t = \frac{t ^{14}}{14}

\int 20 t^{11} d t = \frac{5 t ^{12}}{3}

\int 160 t^{9} d t = 16 t^{10}

\int 640 t^{7} d t = 80 t^{8}

\int 1280 t^{5} d t = \frac{640 t ^{6}}{3}

\int 1024 t^{3} d t = 256 t^{4}

= 3 (\frac{t ^{14}}{14} + \frac{5 t ^{12}}{3} + 16 t^{10} + 80 t^{8} + \frac{640 t ^{6}}{3} + 256 t^{4})

化简

3 (\frac{t ^{14}}{14} + \frac{5 t ^{12}}{3} + 16 t^{10} + 80 t^{8} + \frac{640 t ^{6}}{3} + 256 t^{4}) : \frac{3 t ^{14}}{14} + 5 t^{12} + 48 t^{10} + 240 t^{8} + 640 t^{6} + 768 t^{4}

= \frac{3 t ^{14}}{14} + 5 t^{12} + 48 t^{10} + 240 t^{8} + 640 t^{6} + 768 t^{4}

解答补常数

= \frac{3 t ^{14}}{14} + 5 t^{12} + 48 t^{10} + 240 t^{8} + 640 t^{6} + 768 t^{4} + C

\int \frac{1}{( a - x ) ^{2}} d x

im pl i c i t \frac{d y}{d x}, 2 x y - 4 y = x y - x y^{2}

y^{^{'}} = \frac{9 x ^{2} - 6}{x ^{2}}

a re a 5 x - x^{2}, x, 0, 4

n = 3 \sum \infty \frac{5 ^{n}}{1 4 ^{n}}