integral from x to 5 of sin(t^4)

解

\int_{x}^{5} sin (t^{4}) d t

\frac{5 i ( E _{\frac{3}{4}} ( - 5 ^{4} i ) - E _{\frac{3}{4}} ( 5 ^{4} i ) ) - i x ( E _{\frac{3}{4}} ( - i x ^{4} ) - E _{\frac{3}{4}} ( i x ^{4} ) )}{8}

解答ステップ

\int_{x}^{5} sin (t^{4}) d t

非初等積分を使用する:

\int sin (x^{4}) d x = \frac{1}{8} i x (E_{\frac{3}{4}} (- i x^{4}) - E_{\frac{3}{4}} (i x^{4}))

= [\frac{1}{8} i t (E_{\frac{3}{4}} (- i t^{4}) - E_{\frac{3}{4}} (i t^{4}))]_{x}^{5}

限度を計算する:

i \frac{5}{8} (E_{\frac{3}{4}} (- 5^{4} i) - E_{\frac{3}{4}} (5^{4} i)) - \frac{1}{8} i x (E_{\frac{3}{4}} (- i x^{4}) - E_{\frac{3}{4}} (i x^{4}))

= i \frac{5}{8} (E_{\frac{3}{4}} (- 5^{4} i) - E_{\frac{3}{4}} (5^{4} i)) - \frac{1}{8} i x (E_{\frac{3}{4}} (- i x^{4}) - E_{\frac{3}{4}} (i x^{4}))

簡素化

= \frac{5 i ( E _{\frac{3}{4}} ( - 5 ^{4} i ) - E _{\frac{3}{4}} ( 5 ^{4} i ) ) - i x ( E _{\frac{3}{4}} ( - i x ^{4} ) - E _{\frac{3}{4}} ( i x ^{4} ) )}{8}