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 from-1 to 1 of x^2sqrt(4+x^2)

解

\int_{- 1}^{1} x^{2} 4 + x^{2} d x

解

\frac{- 4 ln ( \frac{3 + 5}{2} ) + 3 5}{2}

+1

十進法表記

1.42925 \dots

解答ステップ

\int_{- 1}^{1} x^{2} 4 + x^{2} d x

三角関数による置換を適用する

= \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} 16 sec^{3} (u) tan^{2} (u) d u

定数を除く:

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

= 16 \cdot \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec^{3} (u) tan^{2} (u) d u

三角関数の公式を使用して書き換える

= 16 \cdot \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec^{3} (u) (- 1 + sec^{2} (u)) d u

拡張

sec^{3} (u) (- 1 + sec^{2} (u)) : - sec^{3} (u) + sec^{5} (u)

= 16 \cdot \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} - sec^{3} (u) + sec^{5} (u) d u

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

= 16 (- \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec^{3} (u) d u + \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec^{5} (u) d u)

\int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec^{3} (u) d u = \frac{5}{4} + \frac{1}{2} ln (\frac{3 + 5}{2})

\int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec^{5} (u) d u = \frac{11 5 + 12 ln ( \frac{3 + 5}{2} )}{32}

= 16 - (\frac{5}{4} + \frac{1}{2} ln (\frac{3 + 5}{2})) + \frac{11 5 + 12 ln ( \frac{3 + 5}{2} )}{32}

簡素化

16 - (\frac{5}{4} + \frac{1}{2} ln (\frac{3 + 5}{2})) + \frac{11 5 + 12 ln ( \frac{3 + 5}{2} )}{32} : \frac{- 4 ln ( \frac{3 + 5}{2} ) + 3 5}{2}

= \frac{- 4 ln ( \frac{3 + 5}{2} ) + 3 5}{2}

グラフ

人気の例

integral from sqrt(x) to x^2 of ln(x)

\int_{x}^{x^{2}} ln (x) d x

integral from x^5 to x^6 of (2t-1)^3

\int_{x^{5}}^{x^{6}} (2 t - 1)^{3} d t

integral from 1 to 3 of (2-3/(x^2))

\int_{1}^{3} (2 - \frac{3}{x ^{2}}) d x

integral from 1 to e of (4(ln(x))^2+x)/x

\int_{1}^{e} \frac{4 ( ln ( x ) ) ^{2} + x}{x} d x

integral from 0 to 1 of x*cos(npix)

\int_{0}^{1} x \cdot cos (nπ x) d x