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 (3(1+t^{1/2})^2)/(2t^{1/2)}

解

\int \frac{3 ( 1 + t ^{\frac{1}{2}} ) ^{2}}{2 t ^{\frac{1}{2}}} d t

解

3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}} + C

解答ステップ

\int \frac{3 ( 1 + t ^{\frac{1}{2}} ) ^{2}}{2 t ^{\frac{1}{2}}} d t

定数を除く:

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

= \frac{3}{2} \cdot \int \frac{( 1 + t ^{\frac{1}{2}} ) ^{2}}{t ^{\frac{1}{2}}} d t

拡張

\frac{( 1 + t ^{\frac{1}{2}} ) ^{2}}{t ^{\frac{1}{2}}} : \frac{1}{t ^{\frac{1}{2}}} + 2 + t^{\frac{1}{2}}

= \frac{3}{2} \cdot \int \frac{1}{t ^{\frac{1}{2}}} + 2 + t^{\frac{1}{2}} d t

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

= \frac{3}{2} (\int \frac{1}{t ^{\frac{1}{2}}} d t + \int 2 d t + \int t^{\frac{1}{2}} d t)

\int \frac{1}{t ^{\frac{1}{2}}} d t = 2 t^{\frac{1}{2}}

\int 2 d t = 2 t

\int t^{\frac{1}{2}} d t = \frac{2}{3} t^{\frac{3}{2}}

= \frac{3}{2} (2 t^{\frac{1}{2}} + 2 t + \frac{2}{3} t^{\frac{3}{2}})

簡素化

\frac{3}{2} (2 t^{\frac{1}{2}} + 2 t + \frac{2}{3} t^{\frac{3}{2}}) : 3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}}

= 3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}}

定数を解答に追加する

= 3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}} + C

グラフ

人気の例

integral of (1+sin^2(x))(1+cos^2(x))

\int (1 + sin^{2} (x)) (1 + cos^{2} (x)) d x

integral of 3/(tan(x))

\int \frac{3}{tan ( x )} d x

integral of ln(tan(x))sec^2(x)

\int ln (tan (x)) sec^{2} (x) d x

integral of sqrt(1/y)

\int \frac{1}{y} d y

integral of xln(x^{-1})

\int x ln (x^{- 1}) d x