What is the integral of e^{7x}cos(6x) ?

The integral of e^{7x}cos(6x) is (6e^{7x}sin(6x))/(85)+(7e^{7x}cos(6x))/(85)+C

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integral of e^{7x}cos(6x)

\int e^{7 x} cos (6 x) d x

\frac{6 e ^{7 x} sin ( 6 x )}{85} + \frac{7 e ^{7 x} cos ( 6 x )}{85} + C

Solution steps

\int e^{7 x} cos (6 x) d x

Apply Integration By Parts

: \frac{1}{6} e^{7 x} sin (6 x) - \int \frac{7}{6} e^{7 x} sin (6 x) d x

= \frac{1}{6} e^{7 x} sin (6 x) - \int \frac{7}{6} e^{7 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^{7 x} sin (6 x) - \frac{7}{6} \cdot \int e^{7 x} sin (6 x) d x

Apply Integration By Parts

: - \frac{1}{6} e^{7 x} cos (6 x) - \int - \frac{7}{6} e^{7 x} cos (6 x) d x

= \frac{1}{6} e^{7 x} sin (6 x) - \frac{7}{6} (- \frac{1}{6} e^{7 x} cos (6 x) - \int - \frac{7}{6} e^{7 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^{7 x} sin (6 x) - \frac{7}{6} (- \frac{1}{6} e^{7 x} cos (6 x) - (- \frac{7}{6} \cdot \int e^{7 x} cos (6 x) d x))

Therefore

\int e^{7 x} cos (6 x) d x = \frac{1}{6} e^{7 x} sin (6 x) - \frac{7}{6} (- \frac{1}{6} e^{7 x} cos (6 x) - (- \frac{7}{6} \cdot \int e^{7 x} cos (6 x) d x))

Isolate

\int e^{7 x} cos (6 x) d x

= \frac{6 e ^{7 x} sin ( 6 x )}{85} + \frac{7 e ^{7 x} cos ( 6 x )}{85}

Add a constant to the solution

= \frac{6 e ^{7 x} sin ( 6 x )}{85} + \frac{7 e ^{7 x} cos ( 6 x )}{85} + C

n \to \infty lim (0. 6^{n})

x \to 0 + lim (2 + 3 e^{- \frac{3}{x}})

\frac{1}{x - 1}

\frac{d}{d x} (x^{π^{2}} + (π^{2})^{x})

\frac{d}{d x} (\frac{sin ( 2 x )}{cos ( 2 x )})

What is the integral of e^{7x}cos(6x) ?
The integral of e^{7x}cos(6x) is (6e^{7x}sin(6x))/(85)+(7e^{7x}cos(6x))/(85)+C