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

The integral of e^{6x}cos(x) is (e^{6x}sin(x))/(37)+(6e^{6x}cos(x))/(37)+C

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

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

\frac{e ^{6 x} sin ( x )}{37} + \frac{6 e ^{6 x} cos ( x )}{37} + C

Solution steps

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

Apply Integration By Parts

: e^{6 x} sin (x) - \int e^{6 x} \cdot 6 sin (x) d x

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

Take the constant out:

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

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

Apply Integration By Parts

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

= e^{6 x} sin (x) - 6 (- e^{6 x} cos (x) - \int - 6 e^{6 x} cos (x) d x)

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{e ^{6 x} sin ( x )}{37} + \frac{6 e ^{6 x} cos ( x )}{37}

Add a constant to the solution

= \frac{e ^{6 x} sin ( x )}{37} + \frac{6 e ^{6 x} cos ( x )}{37} + C

x \to - 3 lim (2 x + 6)

\frac{d}{d x} (x^{52} - \frac{1}{x ^{2}})

\frac{x ^{2} - 1}{2 x - 3}

\int 5 e^{4 x + e^{4 x}} d x

f (x) = (x^{3} + 3 x + 6) (4 + \frac{1}{x ^{2}})

What is the integral of e^{6x}cos(x) ?
The integral of e^{6x}cos(x) is (e^{6x}sin(x))/(37)+(6e^{6x}cos(x))/(37)+C