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受欢迎的微积分 >

integral of 3^xcos(2x)

解答

\int 3^{x} cos (2 x) d x

解答

\frac{2 \cdot 3 ^{x} sin ( 2 x )}{4 + ln ^{2} ( 3 )} + \frac{ln ( 3 ) \cdot 3 ^{x} cos ( 2 x )}{4 + ln ^{2} ( 3 )} + C

求解步骤

\int 3^{x} cos (2 x) d x

使用分布积分法

= 3^{x} \cdot \frac{1}{2} sin (2 x) - \int 3^{x} ln (3) \frac{1}{2} sin (2 x) d x

提出常数:

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

= 3^{x} \cdot \frac{1}{2} sin (2 x) - ln (3) \frac{1}{2} \cdot \int 3^{x} sin (2 x) d x

使用分布积分法

= 3^{x} \cdot \frac{1}{2} sin (2 x) - ln (3) \frac{1}{2} (- \frac{1}{2} \cdot 3^{x} cos (2 x) - \int - \frac{1}{2} ln (3) \cdot 3^{x} cos (2 x) d x)

提出常数:

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

= 3^{x} \cdot \frac{1}{2} sin (2 x) - ln (3) \frac{1}{2} (- \frac{1}{2} \cdot 3^{x} cos (2 x) - (- \frac{1}{2} ln (3) \cdot \int 3^{x} cos (2 x) d x))

因此

\int 3^{x} cos (2 x) d x = 3^{x} \cdot \frac{1}{2} sin (2 x) - ln (3) \frac{1}{2} (- \frac{1}{2} \cdot 3^{x} cos (2 x) - (- \frac{1}{2} ln (3) \cdot \int 3^{x} cos (2 x) d x))

整理

\int 3^{x} cos (2 x) d x

= \frac{2 \cdot 3 ^{x} sin ( 2 x )}{4 + ln ^{2} ( 3 )} + \frac{ln ( 3 ) \cdot 3 ^{x} cos ( 2 x )}{4 + ln ^{2} ( 3 )}

解答补常数

= \frac{2 \cdot 3 ^{x} sin ( 2 x )}{4 + ln ^{2} ( 3 )} + \frac{ln ( 3 ) \cdot 3 ^{x} cos ( 2 x )}{4 + ln ^{2} ( 3 )} + C

作图

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