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人気のある微分積分 >

integral of (6x^8)/(sec(10x^3))

解

\int \frac{6 x ^{8}}{sec ( 10 x ^{3} )} d x

解

\frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3})) + C

解答ステップ

\int \frac{6 x ^{8}}{sec ( 10 x ^{3} )} d x

定数を除く:

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

= 6 \cdot \int \frac{x ^{8}}{sec ( 10 x ^{3} )} d x

u置換積分法を適用する

= 6 \cdot \int \frac{u ^{2}}{3000 sec ( u )} d u

定数を除く:

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

= 6 \cdot \frac{1}{3000} \cdot \int \frac{u ^{2}}{sec ( u )} d u

基本的な三角関数の公式を使用する:

\frac{1}{sec ( x )} = cos (x)

= 6 \cdot \frac{1}{3000} \cdot \int u^{2} cos (u) d u

部分積分を適用する

= 6 \cdot \frac{1}{3000} (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= 6 \cdot \frac{1}{3000} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

代用を戻す

u = 10 x^{3}

= 6 \cdot \frac{1}{3000} ((10 x^{3})^{2} sin (10 x^{3}) - 2 (- 10 x^{3} cos (10 x^{3}) + sin (10 x^{3})))

簡素化

6 \cdot \frac{1}{3000} ((10 x^{3})^{2} sin (10 x^{3}) - 2 (- 10 x^{3} cos (10 x^{3}) + sin (10 x^{3}))) : \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3}))

= \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3}))

定数を解答に追加する

= \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3})) + C

グラフ

人気の例

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