What is the integral of 5^xsin(x) ?

The integral of 5^xsin(x) is -(5^xcos(x))/(1+ln^2(5))+(ln(5)*5^xsin(x))/(1+ln^2(5))+C

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integral of 5^xsin(x)

\int 5^{x} sin (x) d x

- \frac{5 ^{x} cos ( x )}{1 + ln ^{2} ( 5 )} + \frac{ln ( 5 ) \cdot 5 ^{x} sin ( x )}{1 + ln ^{2} ( 5 )} + C

Solution steps

\int 5^{x} sin (x) d x

Apply Integration By Parts

: - 5^{x} cos (x) - \int - ln (5) \cdot 5^{x} cos (x) d x

= - 5^{x} cos (x) - \int - ln (5) \cdot 5^{x} cos (x) d x

Take the constant out:

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

= - 5^{x} cos (x) - (- ln (5) \cdot \int 5^{x} cos (x) d x)

Apply Integration By Parts

: 5^{x} sin (x) - \int 5^{x} ln (5) sin (x) d x

= - 5^{x} cos (x) - (- ln (5) (5^{x} sin (x) - \int 5^{x} ln (5) sin (x) d x))

Take the constant out:

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

= - 5^{x} cos (x) - (- ln (5) (5^{x} sin (x) - ln (5) \cdot \int 5^{x} sin (x) d x))

Therefore

\int 5^{x} sin (x) d x = - 5^{x} cos (x) - (- ln (5) (5^{x} sin (x) - ln (5) \cdot \int 5^{x} sin (x) d x))

Isolate

\int 5^{x} sin (x) d x

= - \frac{5 ^{x} cos ( x )}{1 + ln ^{2} ( 5 )} + \frac{ln ( 5 ) \cdot 5 ^{x} sin ( x )}{1 + ln ^{2} ( 5 )}

Add a constant to the solution

= - \frac{5 ^{x} cos ( x )}{1 + ln ^{2} ( 5 )} + \frac{ln ( 5 ) \cdot 5 ^{x} sin ( x )}{1 + ln ^{2} ( 5 )} + C

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What is the integral of 5^xsin(x) ?
The integral of 5^xsin(x) is -(5^xcos(x))/(1+ln^2(5))+(ln(5)*5^xsin(x))/(1+ln^2(5))+C