What is the integral of e^{3x}cos(x) ?

The integral of e^{3x}cos(x) is (e^{3x}sin(x))/(10)+(3e^{3x}cos(x))/(10)+C

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integral of e^{3x}cos(x)

\int e^{3 x} cos (x) d x

\frac{e ^{3 x} sin ( x )}{10} + \frac{3 e ^{3 x} cos ( x )}{10} + C

Solution steps

\int e^{3 x} cos (x) d x

Apply Integration By Parts

= e^{3 x} sin (x) - \int e^{3 x} \cdot 3 sin (x) d x

Take the constant out:

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

= e^{3 x} sin (x) - 3 \cdot \int e^{3 x} sin (x) d x

Apply Integration By Parts

= e^{3 x} sin (x) - 3 (- e^{3 x} cos (x) - \int - 3 e^{3 x} cos (x) d x)

Take the constant out:

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

= e^{3 x} sin (x) - 3 (- e^{3 x} cos (x) - (- 3 \cdot \int e^{3 x} cos (x) d x))

Therefore

\int e^{3 x} cos (x) d x = e^{3 x} sin (x) - 3 (- e^{3 x} cos (x) - (- 3 \cdot \int e^{3 x} cos (x) d x))

Isolate

\int e^{3 x} cos (x) d x

= \frac{e ^{3 x} sin ( x )}{10} + \frac{3 e ^{3 x} cos ( x )}{10}

Add a constant to the solution

= \frac{e ^{3 x} sin ( x )}{10} + \frac{3 e ^{3 x} cos ( x )}{10} + C

What is the integral of e^{3x}cos(x) ?
The integral of e^{3x}cos(x) is (e^{3x}sin(x))/(10)+(3e^{3x}cos(x))/(10)+C