What is the integral of e^{2z}cos(1/4 z) ?

The integral of e^{2z}cos(1/4 z) is (4e^{2z}sin(1/4 z))/(65)+(32e^{2z}cos(1/4 z))/(65)+C

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integral of e^{2z}cos(1/4 z)

\int e^{2 z} cos (\frac{1}{4} z) d z

\frac{4 e ^{2 z} sin ( \frac{1}{4} z )}{65} + \frac{32 e ^{2 z} cos ( \frac{1}{4} z )}{65} + C

Solution steps

\int e^{2 z} cos (\frac{1}{4} z) d z

Apply Integration By Parts

: 4 e^{2 z} sin (\frac{1}{4} z) - \int 8 e^{2 z} sin (\frac{1}{4} z) d z

= 4 e^{2 z} sin (\frac{1}{4} z) - \int 8 e^{2 z} sin (\frac{1}{4} z) d z

Take the constant out:

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

= 4 e^{2 z} sin (\frac{1}{4} z) - 8 \cdot \int e^{2 z} sin (\frac{1}{4} z) d z

Apply Integration By Parts

: - 4 e^{2 z} cos (\frac{1}{4} z) - \int - 8 e^{2 z} cos (\frac{1}{4} z) d z

= 4 e^{2 z} sin (\frac{1}{4} z) - 8 (- 4 e^{2 z} cos (\frac{1}{4} z) - \int - 8 e^{2 z} cos (\frac{1}{4} z) d z)

Take the constant out:

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

= 4 e^{2 z} sin (\frac{1}{4} z) - 8 (- 4 e^{2 z} cos (\frac{1}{4} z) - (- 8 \cdot \int e^{2 z} cos (\frac{1}{4} z) d z))

Therefore

\int e^{2 z} cos (\frac{1}{4} z) d z = 4 e^{2 z} sin (\frac{1}{4} z) - 8 (- 4 e^{2 z} cos (\frac{1}{4} z) - (- 8 \cdot \int e^{2 z} cos (\frac{1}{4} z) d z))

Isolate

\int e^{2 z} cos (\frac{1}{4} z) d z

= \frac{4 e ^{2 z} sin ( \frac{1}{4} z )}{65} + \frac{32 e ^{2 z} cos ( \frac{1}{4} z )}{65}

Add a constant to the solution

= \frac{4 e ^{2 z} sin ( \frac{1}{4} z )}{65} + \frac{32 e ^{2 z} cos ( \frac{1}{4} z )}{65} + C

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What is the integral of e^{2z}cos(1/4 z) ?
The integral of e^{2z}cos(1/4 z) is (4e^{2z}sin(1/4 z))/(65)+(32e^{2z}cos(1/4 z))/(65)+C