What is the integral of e^{6z}cos(2z) ?

The integral of e^{6z}cos(2z) is (e^{6z}sin(2z))/(20)+(3e^{6z}cos(2z))/(20)+C

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integral of e^{6z}cos(2z)

\int e^{6 z} cos (2 z) d z

\frac{e ^{6 z} sin ( 2 z )}{20} + \frac{3 e ^{6 z} cos ( 2 z )}{20} + C

Solution steps

\int e^{6 z} cos (2 z) d z

Apply Integration By Parts

: \frac{1}{2} e^{6 z} sin (2 z) - \int 3 e^{6 z} sin (2 z) d z

= \frac{1}{2} e^{6 z} sin (2 z) - \int 3 e^{6 z} sin (2 z) d z

Take the constant out:

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

= \frac{1}{2} e^{6 z} sin (2 z) - 3 \cdot \int e^{6 z} sin (2 z) d z

Apply Integration By Parts

: - \frac{1}{2} e^{6 z} cos (2 z) - \int - 3 e^{6 z} cos (2 z) d z

= \frac{1}{2} e^{6 z} sin (2 z) - 3 (- \frac{1}{2} e^{6 z} cos (2 z) - \int - 3 e^{6 z} cos (2 z) d z)

Take the constant out:

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

= \frac{1}{2} e^{6 z} sin (2 z) - 3 (- \frac{1}{2} e^{6 z} cos (2 z) - (- 3 \cdot \int e^{6 z} cos (2 z) d z))

Therefore

\int e^{6 z} cos (2 z) d z = \frac{1}{2} e^{6 z} sin (2 z) - 3 (- \frac{1}{2} e^{6 z} cos (2 z) - (- 3 \cdot \int e^{6 z} cos (2 z) d z))

Isolate

\int e^{6 z} cos (2 z) d z

= \frac{e ^{6 z} sin ( 2 z )}{20} + \frac{3 e ^{6 z} cos ( 2 z )}{20}

Add a constant to the solution

= \frac{e ^{6 z} sin ( 2 z )}{20} + \frac{3 e ^{6 z} cos ( 2 z )}{20} + C

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What is the integral of e^{6z}cos(2z) ?
The integral of e^{6z}cos(2z) is (e^{6z}sin(2z))/(20)+(3e^{6z}cos(2z))/(20)+C