What is the integral of 7e^{7x}sin(7x) ?

The integral of 7e^{7x}sin(7x) is 7(-(e^{7x}cos(7x))/(14)+(e^{7x}sin(7x))/(14))+C

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integral of 7e^{7x}sin(7x)

\int 7 e^{7 x} sin (7 x) d x

7 (- \frac{e ^{7 x} cos ( 7 x )}{14} + \frac{e ^{7 x} sin ( 7 x )}{14}) + C

Solution steps

\int 7 e^{7 x} sin (7 x) d x

Take the constant out:

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

= 7 \cdot \int e^{7 x} sin (7 x) d x

Apply Integration By Parts

= 7 (- \frac{1}{7} e^{7 x} cos (7 x) - \int - e^{7 x} cos (7 x) d x)

Take the constant out:

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

= 7 (- \frac{1}{7} e^{7 x} cos (7 x) - (- \int e^{7 x} cos (7 x) d x))

Apply Integration By Parts

= 7 (- \frac{1}{7} e^{7 x} cos (7 x) - (- (\frac{1}{7} e^{7 x} sin (7 x) - \int e^{7 x} sin (7 x) d x)))

Therefore

7 \cdot \int e^{7 x} sin (7 x) d x = 7 (- \frac{1}{7} e^{7 x} cos (7 x) - (- (\frac{1}{7} e^{7 x} sin (7 x) - \int e^{7 x} sin (7 x) d x)))

Isolate

\int e^{7 x} sin (7 x) d x

= 7 (- \frac{e ^{7 x} cos ( 7 x )}{14} + \frac{e ^{7 x} sin ( 7 x )}{14})

Add a constant to the solution

= 7 (- \frac{e ^{7 x} cos ( 7 x )}{14} + \frac{e ^{7 x} sin ( 7 x )}{14}) + C

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What is the integral of 7e^{7x}sin(7x) ?
The integral of 7e^{7x}sin(7x) is 7(-(e^{7x}cos(7x))/(14)+(e^{7x}sin(7x))/(14))+C