integral of 3x^2ln(x^2+5)

解答

\int 3 x^{2} ln (x^{2} + 5) d x

x^{3} ln (x^{2} + 5) - \frac{2 x ^{3}}{3} + 10 x - 105 arctan (\frac{x}{5}) + C

求解步骤

\int 3 x^{2} ln (x^{2} + 5) d x

提出常数:

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

= 3 \cdot \int x^{2} ln (x^{2} + 5) d x

使用分布积分法

= 3 (\frac{1}{3} x^{3} ln (x^{2} + 5) - \int \frac{2 x ^{4}}{3 ( x ^{2} + 5 )} d x)

\int \frac{2 x ^{4}}{3 ( x ^{2} + 5 )} d x = \frac{2 x ^{3}}{9} - \frac{10 x}{3} + \frac{10 5}{3} arctan (\frac{x}{5})

= 3 (\frac{1}{3} x^{3} ln (x^{2} + 5) - (\frac{2 x ^{3}}{9} - \frac{10 x}{3} + \frac{10 5}{3} arctan (\frac{x}{5})))

化简

3 (\frac{1}{3} x^{3} ln (x^{2} + 5) - (\frac{2 x ^{3}}{9} - \frac{10 x}{3} + \frac{10 5}{3} arctan (\frac{x}{5}))) : x^{3} ln (x^{2} + 5) - \frac{2 x ^{3}}{3} + 10 x - 105 arctan (\frac{x}{5})

= x^{3} ln (x^{2} + 5) - \frac{2 x ^{3}}{3} + 10 x - 105 arctan (\frac{x}{5})

解答补常数

= x^{3} ln (x^{2} + 5) - \frac{2 x ^{3}}{3} + 10 x - 105 arctan (\frac{x}{5}) + C

x \to 0 lim (x [ln (x)]^{2})

\int \frac{( 1 + e ^{x} )}{e ^{x}} d x

x \to 1 lim ((x - 1)^{x - 1})

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