What is the integral of e^{7x}cos(9x) ?

The integral of e^{7x}cos(9x) is (9e^{7x}sin(9x))/(130)+(7e^{7x}cos(9x))/(130)+C

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integral of e^{7x}cos(9x)

\int e^{7 x} cos (9 x) d x

\frac{9 e ^{7 x} sin ( 9 x )}{130} + \frac{7 e ^{7 x} cos ( 9 x )}{130} + C

Solution steps

\int e^{7 x} cos (9 x) d x

Apply Integration By Parts

: \frac{1}{9} e^{7 x} sin (9 x) - \int \frac{7}{9} e^{7 x} sin (9 x) d x

= \frac{1}{9} e^{7 x} sin (9 x) - \int \frac{7}{9} e^{7 x} sin (9 x) d x

Take the constant out:

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

= \frac{1}{9} e^{7 x} sin (9 x) - \frac{7}{9} \cdot \int e^{7 x} sin (9 x) d x

Apply Integration By Parts

: - \frac{1}{9} e^{7 x} cos (9 x) - \int - \frac{7}{9} e^{7 x} cos (9 x) d x

= \frac{1}{9} e^{7 x} sin (9 x) - \frac{7}{9} (- \frac{1}{9} e^{7 x} cos (9 x) - \int - \frac{7}{9} e^{7 x} cos (9 x) d x)

Take the constant out:

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

= \frac{1}{9} e^{7 x} sin (9 x) - \frac{7}{9} (- \frac{1}{9} e^{7 x} cos (9 x) - (- \frac{7}{9} \cdot \int e^{7 x} cos (9 x) d x))

Therefore

\int e^{7 x} cos (9 x) d x = \frac{1}{9} e^{7 x} sin (9 x) - \frac{7}{9} (- \frac{1}{9} e^{7 x} cos (9 x) - (- \frac{7}{9} \cdot \int e^{7 x} cos (9 x) d x))

Isolate

\int e^{7 x} cos (9 x) d x

= \frac{9 e ^{7 x} sin ( 9 x )}{130} + \frac{7 e ^{7 x} cos ( 9 x )}{130}

Add a constant to the solution

= \frac{9 e ^{7 x} sin ( 9 x )}{130} + \frac{7 e ^{7 x} cos ( 9 x )}{130} + C

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\int \frac{25}{x} d x

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\frac{d}{d x} (\frac{( x ^{5} + 3 x ) ^{4}}{cos ( x )})

What is the integral of e^{7x}cos(9x) ?
The integral of e^{7x}cos(9x) is (9e^{7x}sin(9x))/(130)+(7e^{7x}cos(9x))/(130)+C