integral of (4x^2+7)^2

解答

\int (4 x^{2} + 7)^{2} d x

\frac{16 x ^{5}}{5} + \frac{56 x ^{3}}{3} + 49 x + C

求解步骤

\int (4 x^{2} + 7)^{2} d x

乘开

(4 x^{2} + 7)^{2} : 16 x^{4} + 56 x^{2} + 49

= \int 16 x^{4} + 56 x^{2} + 49 d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 16 x^{4} d x + \int 56 x^{2} d x + \int 49 d x

\int 16 x^{4} d x = \frac{16 x ^{5}}{5}

\int 56 x^{2} d x = \frac{56 x ^{3}}{3}

\int 49 d x = 49 x

= \frac{16 x ^{5}}{5} + \frac{56 x ^{3}}{3} + 49 x

解答补常数

= \frac{16 x ^{5}}{5} + \frac{56 x ^{3}}{3} + 49 x + C

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