integral of (8+4x)/(49+64x^2)

解

\int \frac{8 + 4 x}{49 + 64 x ^{2}} d x

\frac{1}{7} arctan (\frac{8}{7} x) + \frac{1}{32} ln 49 + 64 x^{2} + C

解答ステップ

\int \frac{8 + 4 x}{49 + 64 x ^{2}} d x

拡張

\frac{8 + 4 x}{49 + 64 x ^{2}} : \frac{8}{49 + 64 x ^{2}} + \frac{4 x}{49 + 64 x ^{2}}

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

= \int \frac{8}{49 + 64 x ^{2}} d x + \int \frac{4 x}{49 + 64 x ^{2}} d x

\int \frac{8}{49 + 64 x ^{2}} d x = \frac{1}{7} arctan (\frac{8}{7} x)

\int \frac{4 x}{49 + 64 x ^{2}} d x = \frac{1}{32} ln 49 + 64 x^{2}

= \frac{1}{7} arctan (\frac{8}{7} x) + \frac{1}{32} ln 49 + 64 x^{2}

定数を解答に追加する

= \frac{1}{7} arctan (\frac{8}{7} x) + \frac{1}{32} ln 49 + 64 x^{2} + C