integral of 2xarcsin(x^2)

解答

\int 2 x arcsin (x^{2}) d x

x^{2} arcsin (x^{2}) + 1 - x^{4} + C

求解步骤

\int 2 x arcsin (x^{2}) d x

提出常数:

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

= 2 \cdot \int x arcsin (x^{2}) d x

使用分布积分法

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

\int \frac{x ^{3}}{1 - x ^{4}} d x = - \frac{1}{2} 1 - x^{4}

= 2 (\frac{1}{2} x^{2} arcsin (x^{2}) - (- \frac{1}{2} 1 - x^{4}))

化简

2 (\frac{1}{2} x^{2} arcsin (x^{2}) - (- \frac{1}{2} 1 - x^{4})) : x^{2} arcsin (x^{2}) + 1 - x^{4}

= x^{2} arcsin (x^{2}) + 1 - x^{4}

解答补常数

= x^{2} arcsin (x^{2}) + 1 - x^{4} + C

\int \frac{arctan ( x )}{x ^{2} + 1} d x

\int \frac{( x + 1 )}{( x ^{2} - 2 x + 5 )} d x

\int e^{yz} d x

\int \frac{x + x ^{2}}{x} d x

\int (2 - x) x d x