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人気のある微分積分 >

integral from 3 to 5 of sqrt(3+x^2)

解

\int_{3}^{5} 3 + x^{2} d x

解

57 - 33 + \frac{3}{2} ln (\frac{5 3 + 2 21}{3}) - \frac{3}{2} ln (3 + 2)

+1

十進法表記

8.73018 \dots

解答ステップ

\int_{3}^{5} 3 + x^{2} d x

三角関数による置換を適用する

= \int_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} 3 sec^{3} (u) d u

定数を除く:

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

= 3 \cdot \int_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} sec^{3} (u) d u

積分換算を適用する

= 3 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} + \frac{1}{2} \cdot \int_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} sec (u) d u

\int_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} sec (u) d u = ln (\frac{5 3 + 2 21}{3}) - ln (3 + 2)

= 3 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} + \frac{1}{2} (ln (\frac{5 3 + 2 21}{3}) - ln (3 + 2))

簡素化

3 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} + \frac{1}{2} (ln (\frac{5 3 + 2 21}{3}) - ln (3 + 2)) : 3 [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} + \frac{3}{2} ln (\frac{5 3 + 2 21}{3}) - \frac{3}{2} ln (3 + 2)

= 3 [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{3}}^{a r c t a n (\frac{5}{3})} + \frac{3}{2} ln (\frac{5 3 + 2 21}{3}) - \frac{3}{2} ln (3 + 2)

限度を計算する:

\frac{5 7}{3} - 3

= 3 (\frac{5 7}{3} - 3) + \frac{3}{2} ln (\frac{5 3 + 2 21}{3}) - \frac{3}{2} ln (3 + 2)

簡素化

= 57 - 33 + \frac{3}{2} ln (\frac{5 3 + 2 21}{3}) - \frac{3}{2} ln (3 + 2)

グラフ

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