What is the integral of e^{5x}cos(6x) ?

The integral of e^{5x}cos(6x) is (6e^{5x}sin(6x))/(61)+(5e^{5x}cos(6x))/(61)+C

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integral of e^{5x}cos(6x)

\int e^{5 x} cos (6 x) d x

\frac{6 e ^{5 x} sin ( 6 x )}{61} + \frac{5 e ^{5 x} cos ( 6 x )}{61} + C

Solution steps

\int e^{5 x} cos (6 x) d x

Apply Integration By Parts

: \frac{1}{6} e^{5 x} sin (6 x) - \int \frac{5}{6} e^{5 x} sin (6 x) d x

= \frac{1}{6} e^{5 x} sin (6 x) - \int \frac{5}{6} e^{5 x} sin (6 x) d x

Take the constant out:

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

= \frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} \cdot \int e^{5 x} sin (6 x) d x

Apply Integration By Parts

: - \frac{1}{6} e^{5 x} cos (6 x) - \int - \frac{5}{6} e^{5 x} cos (6 x) d x

= \frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} (- \frac{1}{6} e^{5 x} cos (6 x) - \int - \frac{5}{6} e^{5 x} cos (6 x) d x)

Take the constant out:

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

= \frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} (- \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} \cdot \int e^{5 x} cos (6 x) d x))

Therefore

\int e^{5 x} cos (6 x) d x = \frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} (- \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} \cdot \int e^{5 x} cos (6 x) d x))

Isolate

\int e^{5 x} cos (6 x) d x

= \frac{6 e ^{5 x} sin ( 6 x )}{61} + \frac{5 e ^{5 x} cos ( 6 x )}{61}

Add a constant to the solution

= \frac{6 e ^{5 x} sin ( 6 x )}{61} + \frac{5 e ^{5 x} cos ( 6 x )}{61} + C

\frac{d}{d x} ((ln (x^{2}))^{- 3})

\frac{d}{d x} (2 sin (π) x + 3 cos (π) x)

f (x) = \frac{7}{1 - x}

x \to 1.999 lim (2 x + 3)

\frac{d y}{d x} = (3 y + 6 x + 6 - 2)

What is the integral of e^{5x}cos(6x) ?
The integral of e^{5x}cos(6x) is (6e^{5x}sin(6x))/(61)+(5e^{5x}cos(6x))/(61)+C