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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

x \to 1 - lim (x^{2})

x (x - 3)^{2}

\int (\frac{x}{1 + x ^{2}}) d x

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

x \to 5 lim (\frac{3 x + 3}{x - 5})

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