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人気のある微分積分 >

integral of 4^{ex}sin(2)(x)

解

\int 4^{e x} sin (2) (x) d x

解

sin (2) \frac{1}{e ^{2}} (\frac{2 ^{2 e x - 1} e x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{2 e x - 2}) + C

解答ステップ

\int 4^{e x} sin (2) x d x

定数を除く:

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

= sin (2) \cdot \int 4^{e x} x d x

u置換積分法を適用する

= sin (2) \cdot \int \frac{4 ^{u} u}{e ^{2}} d u

定数を除く:

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

= sin (2) \frac{1}{e ^{2}} \cdot \int 4^{u} u d u

部分積分を適用する

= sin (2) \frac{1}{e ^{2}} (\frac{2 ^{2 u - 1} u}{ln ( 2 )} - \int \frac{2 ^{2 u - 1}}{ln ( 2 )} d u)

\int \frac{2 ^{2 u - 1}}{ln ( 2 )} d u = \frac{1}{ln ^{2} ( 2 )} \cdot 2^{2 u - 2}

= sin (2) \frac{1}{e ^{2}} (\frac{2 ^{2 u - 1} u}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{2 u - 2})

代用を戻す

u = e x

= sin (2) \frac{1}{e ^{2}} (\frac{2 ^{2 e x - 1} e x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{2 e x - 2})

定数を解答に追加する

= sin (2) \frac{1}{e ^{2}} (\frac{2 ^{2 e x - 1} e x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{2 e x - 2}) + C

グラフ

人気の例

derivative f(x)=(x^3+2)(3x^2+1)

d er i v a t i v e f (x) = (x^{3} + 2) (3 x^{2} + 1)

integral of (x^3)/((3-x^4)^2)

\int \frac{x ^{3}}{( 3 - x ^{4} ) ^{2}} d x

derivative y=2sin(4x)

d er i v a t i v e y = 2 sin (4 x)

y^'-8/x y=(y^3)/(x^{11)}

y^{^{'}} - \frac{8}{x} y = \frac{y ^{3}}{x ^{11}}

limit as x approaches 3-of |x-3|

x \to 3 - lim (∣ x - 3 ∣)