ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of (2^{2x}-3^x)^2

解

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

解

\frac{2 ^{4 x - 2}}{ln ( 2 )} + \frac{3 ^{x} \cdot 2 ^{2 x + 1}}{ln ( 12 )} - \frac{3 ^{x} \cdot 2 ^{2 x + 2}}{ln ( 12 )} + \frac{9 ^{x}}{2 ln ( 3 )} + C

解答ステップ

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

因数

2^{2 x} - 3^{x} : - (- 2^{2 x} + 3^{x})

= \int (- (- 2^{2 x} + 3^{x}))^{2} d x

簡素化

(- (- 2^{2 x} + 3^{x}))^{2} : (- 4^{x} + 3^{x})^{2}

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

拡張

(- 4^{x} + 3^{x})^{2} : 1 6^{x} - 3^{x} \cdot 2^{1 + 2 x} + 9^{x}

= \int 1 6^{x} - 3^{x} \cdot 2^{1 + 2 x} + 9^{x} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 6^{x} d x - \int 3^{x} \cdot 2^{1 + 2 x} d x + \int 9^{x} d x

\int 1 6^{x} d x = \frac{2 ^{4 x - 2}}{ln ( 2 )}

\int 3^{x} \cdot 2^{1 + 2 x} d x = - \frac{3 ^{x} \cdot 2 ^{2 x + 1}}{ln ( 12 )} + \frac{3 ^{x} \cdot 2 ^{2 x + 2}}{ln ( 12 )}

\int 9^{x} d x = \frac{9 ^{x}}{2 ln ( 3 )}

= \frac{2 ^{4 x - 2}}{ln ( 2 )} - (- \frac{3 ^{x} \cdot 2 ^{2 x + 1}}{ln ( 12 )} + \frac{3 ^{x} \cdot 2 ^{2 x + 2}}{ln ( 12 )}) + \frac{9 ^{x}}{2 ln ( 3 )}

簡素化

= \frac{2 ^{4 x - 2}}{ln ( 2 )} + \frac{3 ^{x} \cdot 2 ^{2 x + 1}}{ln ( 12 )} - \frac{3 ^{x} \cdot 2 ^{2 x + 2}}{ln ( 12 )} + \frac{9 ^{x}}{2 ln ( 3 )}

定数を解答に追加する

= \frac{2 ^{4 x - 2}}{ln ( 2 )} + \frac{3 ^{x} \cdot 2 ^{2 x + 1}}{ln ( 12 )} - \frac{3 ^{x} \cdot 2 ^{2 x + 2}}{ln ( 12 )} + \frac{9 ^{x}}{2 ln ( 3 )} + C

グラフ

人気の例

integral of (2x^2-3x+1)/(x^3-6x^2+8x)

\int \frac{2 x ^{2} - 3 x + 1}{x ^{3} - 6 x ^{2} + 8 x} d x

integral of (xe^x)/(x+1)

\int \frac{x e ^{x}}{x + 1} d x

integral of sin(2pix)sin(npix)

\int sin (2 π x) sin (nπ x) d x

integral of (x+7)/(x^2+4x+4)

\int \frac{x + 7}{x ^{2} + 4 x + 4} d x

integral of ((x^2-x+6))/((x^3+3x))

\int \frac{( x ^{2} - x + 6 )}{( x ^{3} + 3 x )} d x