integral from 0 to 2 of 3xsqrt(x^2+4a^2)

해법

\int_{0}^{2} 3 x x^{2} + 4 a^{2} d x

8 ((a^{2} + 1)^{\frac{3}{2}} - a^{3})

솔루션 단계

\int_{0}^{2} 3 x x^{2} + 4 a^{2} d x

정수를 빼라:

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

= 3 \cdot \int_{0}^{2} x x^{2} + 4 a^{2} d x

대체 적용

: \int_{4 a^{2}}^{4 + 4 a^{2}} \frac{u}{2} d u

= 3 \cdot \int_{4 a^{2}}^{4 + 4 a^{2}} \frac{u}{2} d u

정수를 빼라:

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

= 3 \cdot \frac{1}{2} \cdot \int_{4 a^{2}}^{4 + 4 a^{2}} u d u

전원 규칙 적용

: [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

= 3 \cdot \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

3 \cdot \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

간소화하다 :

\frac{3}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

= \frac{3}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

경계 계산:

\frac{16}{3} (1 + a^{2})^{\frac{3}{2}} - \frac{16}{3} a^{3}

= \frac{3}{2} (\frac{16}{3} (1 + a^{2})^{\frac{3}{2}} - \frac{16}{3} a^{3})

단순화

= 8 ((a^{2} + 1)^{\frac{3}{2}} - a^{3})