What is the integral of e^{4x}cos(5x) ?

The integral of e^{4x}cos(5x) is (5e^{4x}sin(5x))/(41)+(4e^{4x}cos(5x))/(41)+C

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integral of e^{4x}cos(5x)

\int e^{4 x} cos (5 x) d x

\frac{5 e ^{4 x} sin ( 5 x )}{41} + \frac{4 e ^{4 x} cos ( 5 x )}{41} + C

Solution steps

\int e^{4 x} cos (5 x) d x

Apply Integration By Parts

: \frac{1}{5} e^{4 x} sin (5 x) - \int \frac{4}{5} e^{4 x} sin (5 x) d x

= \frac{1}{5} e^{4 x} sin (5 x) - \int \frac{4}{5} e^{4 x} sin (5 x) d x

Take the constant out:

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

= \frac{1}{5} e^{4 x} sin (5 x) - \frac{4}{5} \cdot \int e^{4 x} sin (5 x) d x

Apply Integration By Parts

: - \frac{1}{5} e^{4 x} cos (5 x) - \int - \frac{4}{5} e^{4 x} cos (5 x) d x

= \frac{1}{5} e^{4 x} sin (5 x) - \frac{4}{5} (- \frac{1}{5} e^{4 x} cos (5 x) - \int - \frac{4}{5} e^{4 x} cos (5 x) d x)

Take the constant out:

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

= \frac{1}{5} e^{4 x} sin (5 x) - \frac{4}{5} (- \frac{1}{5} e^{4 x} cos (5 x) - (- \frac{4}{5} \cdot \int e^{4 x} cos (5 x) d x))

Therefore

\int e^{4 x} cos (5 x) d x = \frac{1}{5} e^{4 x} sin (5 x) - \frac{4}{5} (- \frac{1}{5} e^{4 x} cos (5 x) - (- \frac{4}{5} \cdot \int e^{4 x} cos (5 x) d x))

Isolate

\int e^{4 x} cos (5 x) d x

= \frac{5 e ^{4 x} sin ( 5 x )}{41} + \frac{4 e ^{4 x} cos ( 5 x )}{41}

Add a constant to the solution

= \frac{5 e ^{4 x} sin ( 5 x )}{41} + \frac{4 e ^{4 x} cos ( 5 x )}{41} + C

\frac{1}{14 x ^{2}}, \frac{x}{4}, 0, 3 \frac{1}{14}

x \to 2 + lim (\frac{- 4 - x}{7 x - 2})

\int 3 x^{2} - 2 x + 3 d x

\frac{d}{d x} (- \frac{1}{3} x^{3})

y^{^{'}} - k y = a + b e^{- k t}, y (0) = 0

What is the integral of e^{4x}cos(5x) ?
The integral of e^{4x}cos(5x) is (5e^{4x}sin(5x))/(41)+(4e^{4x}cos(5x))/(41)+C