What is the integral of e^{8x}cos(7x) ?

The integral of e^{8x}cos(7x) is (7e^{8x}sin(7x))/(113)+(8e^{8x}cos(7x))/(113)+C

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integral of e^{8x}cos(7x)

\int e^{8 x} cos (7 x) d x

\frac{7 e ^{8 x} sin ( 7 x )}{113} + \frac{8 e ^{8 x} cos ( 7 x )}{113} + C

Solution steps

\int e^{8 x} cos (7 x) d x

Apply Integration By Parts

: \frac{1}{7} e^{8 x} sin (7 x) - \int \frac{8}{7} e^{8 x} sin (7 x) d x

= \frac{1}{7} e^{8 x} sin (7 x) - \int \frac{8}{7} e^{8 x} sin (7 x) d x

Take the constant out:

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

= \frac{1}{7} e^{8 x} sin (7 x) - \frac{8}{7} \cdot \int e^{8 x} sin (7 x) d x

Apply Integration By Parts

: - \frac{1}{7} e^{8 x} cos (7 x) - \int - \frac{8}{7} e^{8 x} cos (7 x) d x

= \frac{1}{7} e^{8 x} sin (7 x) - \frac{8}{7} (- \frac{1}{7} e^{8 x} cos (7 x) - \int - \frac{8}{7} e^{8 x} cos (7 x) d x)

Take the constant out:

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

= \frac{1}{7} e^{8 x} sin (7 x) - \frac{8}{7} (- \frac{1}{7} e^{8 x} cos (7 x) - (- \frac{8}{7} \cdot \int e^{8 x} cos (7 x) d x))

Therefore

\int e^{8 x} cos (7 x) d x = \frac{1}{7} e^{8 x} sin (7 x) - \frac{8}{7} (- \frac{1}{7} e^{8 x} cos (7 x) - (- \frac{8}{7} \cdot \int e^{8 x} cos (7 x) d x))

Isolate

\int e^{8 x} cos (7 x) d x

= \frac{7 e ^{8 x} sin ( 7 x )}{113} + \frac{8 e ^{8 x} cos ( 7 x )}{113}

Add a constant to the solution

= \frac{7 e ^{8 x} sin ( 7 x )}{113} + \frac{8 e ^{8 x} cos ( 7 x )}{113} + C

\int x^{12} d x

x \to 2 - lim (∣ x - 2 ∣)

\frac{\partial}{\partial x} (e^{- x} sin (x + y))

\frac{d}{d x} (0.5 x^{2})

\frac{\partial}{\partial y} (\frac{y}{x ^{2}})

What is the integral of e^{8x}cos(7x) ?
The integral of e^{8x}cos(7x) is (7e^{8x}sin(7x))/(113)+(8e^{8x}cos(7x))/(113)+C