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受欢迎的微积分 >

integral of 2xln(x^2+1)

解答

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

解答

x^{2} ln (x^{2} + 1) + ln (x^{2} + 1) - x^{2} - 1 + C

求解步骤

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

提出常数:

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

= 2 \cdot \int x ln (x^{2} + 1) d x

使用换元积分法

= 2 \cdot \int \frac{ln ( u )}{2} d u

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int ln (u) d u

使用分布积分法

= 2 \cdot \frac{1}{2} (u ln (u) - \int 1 d u)

\int 1 d u = u

= 2 \cdot \frac{1}{2} (u ln (u) - u)

u = x^{2} + 1

代回

= 2 \cdot \frac{1}{2} ((x^{2} + 1) ln (x^{2} + 1) - (x^{2} + 1))

化简

2 \cdot \frac{1}{2} ((x^{2} + 1) ln (x^{2} + 1) - (x^{2} + 1)) : x^{2} ln (x^{2} + 1) + ln (x^{2} + 1) - x^{2} - 1

= x^{2} ln (x^{2} + 1) + ln (x^{2} + 1) - x^{2} - 1

解答补常数

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

作图

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