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人気のある微分積分 >

integral of sqrt((3z+1)^2-2)

解

\int (3 z + 1)^{2} - 2 d z

解

\frac{1}{6} (- 2 ln 2 \frac{9 z ^{2} + 6 z - 1}{2} + 3 z + 1 + ln (2) + 3 z 9 z^{2} + 6 z - 1 + 9 z^{2} + 6 z - 1) + C

解答ステップ

\int (3 z + 1)^{2} - 2 d z

u置換積分法を適用する

= \int \frac{u ^{2} - 2}{3} d u

定数を除く:

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

= \frac{1}{3} \cdot \int u^{2} - 2 d u

三角関数による置換を適用する

= \frac{1}{3} \cdot \int 2 tan^{2} (v) sec (v) d v

定数を除く:

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

= \frac{1}{3} \cdot 2 \cdot \int tan^{2} (v) sec (v) d v

三角関数の公式を使用して書き換える

= \frac{1}{3} \cdot 2 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

拡張

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

= \frac{1}{3} \cdot 2 \cdot \int - sec (v) + sec^{3} (v) d v

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{1}{3} \cdot 2 (- \int sec (v) d v + \int sec^{3} (v) d v)

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

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

= \frac{1}{3} \cdot 2 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代用を戻す

= \frac{1}{3} \cdot 2 (- ln tan (arcsec (\frac{1}{2} (3 z + 1))) + sec (arcsec (\frac{1}{2} (3 z + 1))) + \frac{1}{2} sec (arcsec (\frac{1}{2} (3 z + 1))) tan (arcsec (\frac{1}{2} (3 z + 1))) + \frac{1}{2} ln tan (arcsec (\frac{1}{2} (3 z + 1))) + sec (arcsec (\frac{1}{2} (3 z + 1))))

簡素化

\frac{1}{3} \cdot 2 (- ln tan (arcsec (\frac{1}{2} (3 z + 1))) + sec (arcsec (\frac{1}{2} (3 z + 1))) + \frac{1}{2} sec (arcsec (\frac{1}{2} (3 z + 1))) tan (arcsec (\frac{1}{2} (3 z + 1))) + \frac{1}{2} ln tan (arcsec (\frac{1}{2} (3 z + 1))) + sec (arcsec (\frac{1}{2} (3 z + 1)))) : \frac{1}{6} (- 2 ln 2 \frac{9 z ^{2} + 6 z - 1}{2} + 3 z + 1 + ln (2) + 3 z 9 z^{2} + 6 z - 1 + 9 z^{2} + 6 z - 1)

= \frac{1}{6} (- 2 ln 2 \frac{9 z ^{2} + 6 z - 1}{2} + 3 z + 1 + ln (2) + 3 z 9 z^{2} + 6 z - 1 + 9 z^{2} + 6 z - 1)

定数を解答に追加する

= \frac{1}{6} (- 2 ln 2 \frac{9 z ^{2} + 6 z - 1}{2} + 3 z + 1 + ln (2) + 3 z 9 z^{2} + 6 z - 1 + 9 z^{2} + 6 z - 1) + C

グラフ

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