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

The integral of 1/2 e^{2x}cos(x) is 1/10 (e^{2x}sin(x)+2e^{2x}cos(x))+C

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

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

\frac{1}{10} (e^{2 x} sin (x) + 2 e^{2 x} cos (x)) + C

Solution steps

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

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{2 x} cos (x) d x

Apply Integration By Parts

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

Take the constant out:

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

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

Apply Integration By Parts

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

Take the constant out:

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

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

Therefore

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

Isolate

\int e^{2 x} cos (x) d x

= \frac{1}{2} (\frac{e ^{2 x} sin ( x )}{5} + \frac{2 e ^{2 x} cos ( x )}{5})

Simplify

\frac{1}{2} (\frac{e ^{2 x} sin ( x )}{5} + \frac{2 e ^{2 x} cos ( x )}{5}) : \frac{1}{10} (e^{2 x} sin (x) + 2 e^{2 x} cos (x))

= \frac{1}{10} (e^{2 x} sin (x) + 2 e^{2 x} cos (x))

Add a constant to the solution

= \frac{1}{10} (e^{2 x} sin (x) + 2 e^{2 x} cos (x)) + C

\frac{\partial ^{2}}{\partial x \partial y} (sin^{2} (m x + n y))

d er i v a t i v e q (t) = sin^{2} (\frac{t ^{4}}{t ^{5} - 6 t})

\int_{1}^{4} 1 t ln (t) d t

\frac{d x}{d t} = \frac{1}{x e ^{t + 3 x}}

a re a 3 x^{2} + 8 x + 2, 2 x^{2} + 4 x - 1

What is the integral of 1/2 e^{2x}cos(x) ?
The integral of 1/2 e^{2x}cos(x) is 1/10 (e^{2x}sin(x)+2e^{2x}cos(x))+C