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 (9+16x^2)^{3/2}

解

\int (9 + 16 x^{2})^{\frac{3}{2}} d x

解

\frac{1}{32} 8 x (16 x^{2} + 9)^{\frac{3}{2}} + 27 4 x (16 x^{2} + 9)^{\frac{1}{2}} + 9 ln \frac{4 x + ( 16 x ^{2} + 9 ) ^{\frac{1}{2}}}{3} + C

解答ステップ

\int (9 + 16 x^{2})^{\frac{3}{2}} d x

三角関数による置換を適用する

= \int \frac{81}{4} sec^{5} (u) d u

定数を除く:

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

= \frac{81}{4} \cdot \int sec^{5} (u) d u

積分換算を適用する

= \frac{81}{4} (\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} \cdot \int sec^{3} (u) d u)

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= \frac{81}{4} (\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣))

代用を戻す

u = arctan (\frac{4}{3} x)

= \frac{81}{4} (\frac{sec ^{4} ( arctan ( \frac{4}{3} x ) ) sin ( arctan ( \frac{4}{3} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{4}{3} x)) tan (arctan (\frac{4}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{4}{3} x)) + sec (arctan (\frac{4}{3} x))))

簡素化

\frac{81}{4} (\frac{sec ^{4} ( arctan ( \frac{4}{3} x ) ) sin ( arctan ( \frac{4}{3} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{4}{3} x)) tan (arctan (\frac{4}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{4}{3} x)) + sec (arctan (\frac{4}{3} x)))) : \frac{1}{32} 8 x (16 x^{2} + 9)^{\frac{3}{2}} + 27 4 x (16 x^{2} + 9)^{\frac{1}{2}} + 9 ln \frac{4 x + ( 16 x ^{2} + 9 ) ^{\frac{1}{2}}}{3}

= \frac{1}{32} 8 x (16 x^{2} + 9)^{\frac{3}{2}} + 27 4 x (16 x^{2} + 9)^{\frac{1}{2}} + 9 ln \frac{4 x + ( 16 x ^{2} + 9 ) ^{\frac{1}{2}}}{3}

定数を解答に追加する

= \frac{1}{32} 8 x (16 x^{2} + 9)^{\frac{3}{2}} + 27 4 x (16 x^{2} + 9)^{\frac{1}{2}} + 9 ln \frac{4 x + ( 16 x ^{2} + 9 ) ^{\frac{1}{2}}}{3} + C

グラフ

人気の例

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

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

integral of x/(\sqrt[3]{x^2+3)}

\int \frac{x}{3 x ^{2} + 3} d x

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

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

integral of 1/(sqrt(1-x^2)*arcsin(x))

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

integral of (x^2-5sin(x)+sec^2(x))

\int (x^{2} - 5 sin (x) + sec^{2} (x)) d x