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 (t^2)/(sqrt(a^4+t^2))

解

\int \frac{t ^{2}}{a ^{4} + t ^{2}} d t

解

\frac{1}{2} t t^{2} + a^{4} - \frac{a ^{4}}{2} (ln t + t^{2} + a^{4} - 2 ln (a)) + C

解答ステップ

\int \frac{t ^{2}}{a ^{4} + t ^{2}} d t

三角関数による置換を適用する

= \int a^{4} tan^{2} (u) sec (u) d u

定数を除く:

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

= a^{4} \cdot \int tan^{2} (u) sec (u) d u

三角関数の公式を使用して書き換える

= a^{4} \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

拡張

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

= a^{4} \cdot \int - sec (u) + sec^{3} (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

= a^{4} (- \int sec (u) d u + \int sec^{3} (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

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

= a^{4} (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

代用を戻す

u = arctan (\frac{1}{a ^{2}} t)

= a^{4} (- ln tan (arctan (\frac{1}{a ^{2}} t)) + sec (arctan (\frac{1}{a ^{2}} t)) + \frac{1}{2} sec (arctan (\frac{1}{a ^{2}} t)) tan (arctan (\frac{1}{a ^{2}} t)) + \frac{1}{2} ln tan (arctan (\frac{1}{a ^{2}} t)) + sec (arctan (\frac{1}{a ^{2}} t)))

簡素化

a^{4} (- ln tan (arctan (\frac{1}{a ^{2}} t)) + sec (arctan (\frac{1}{a ^{2}} t)) + \frac{1}{2} sec (arctan (\frac{1}{a ^{2}} t)) tan (arctan (\frac{1}{a ^{2}} t)) + \frac{1}{2} ln tan (arctan (\frac{1}{a ^{2}} t)) + sec (arctan (\frac{1}{a ^{2}} t))) : \frac{1}{2} t t^{2} + a^{4} - \frac{a ^{4}}{2} (ln t + t^{2} + a^{4} - 2 ln (a))

= \frac{1}{2} t t^{2} + a^{4} - \frac{a ^{4}}{2} (ln t + t^{2} + a^{4} - 2 ln (a))

定数を解答に追加する

= \frac{1}{2} t t^{2} + a^{4} - \frac{a ^{4}}{2} (ln t + t^{2} + a^{4} - 2 ln (a)) + C

グラフ

人気の例

integral of (5/(x^2))

\int (\frac{5}{x ^{2}}) d x

derivative f(x)=(x^2-4)*(x+4)

d er i v a t i v e f (x) = (x^{2} - 4) \cdot (x + 4)

derivative of (5sqrt(x)/6)

\frac{d}{d x} (\frac{5 x}{6})

integral of 2^x+4sinh(x)

\int 2^{x} + 4 sinh (x) d x

derivative f(x)= 2/(sqrt(6+8x))

d er i v a t i v e f (x) = \frac{2}{6 + 8 x}