What is the integral of e^{5x}cos(7x) ?

The integral of e^{5x}cos(7x) is (7e^{5x}sin(7x))/(74)+(5e^{5x}cos(7x))/(74)+C

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integral of e^{5x}cos(7x)

\int e^{5 x} cos (7 x) d x

\frac{7 e ^{5 x} sin ( 7 x )}{74} + \frac{5 e ^{5 x} cos ( 7 x )}{74} + C

Solution steps

\int e^{5 x} cos (7 x) d x

Apply Integration By Parts

: \frac{1}{7} e^{5 x} sin (7 x) - \int \frac{5}{7} e^{5 x} sin (7 x) d x

= \frac{1}{7} e^{5 x} sin (7 x) - \int \frac{5}{7} e^{5 x} sin (7 x) d x

Take the constant out:

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

= \frac{1}{7} e^{5 x} sin (7 x) - \frac{5}{7} \cdot \int e^{5 x} sin (7 x) d x

Apply Integration By Parts

: - \frac{1}{7} e^{5 x} cos (7 x) - \int - \frac{5}{7} e^{5 x} cos (7 x) d x

= \frac{1}{7} e^{5 x} sin (7 x) - \frac{5}{7} (- \frac{1}{7} e^{5 x} cos (7 x) - \int - \frac{5}{7} e^{5 x} cos (7 x) d x)

Take the constant out:

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

= \frac{1}{7} e^{5 x} sin (7 x) - \frac{5}{7} (- \frac{1}{7} e^{5 x} cos (7 x) - (- \frac{5}{7} \cdot \int e^{5 x} cos (7 x) d x))

Therefore

\int e^{5 x} cos (7 x) d x = \frac{1}{7} e^{5 x} sin (7 x) - \frac{5}{7} (- \frac{1}{7} e^{5 x} cos (7 x) - (- \frac{5}{7} \cdot \int e^{5 x} cos (7 x) d x))

Isolate

\int e^{5 x} cos (7 x) d x

= \frac{7 e ^{5 x} sin ( 7 x )}{74} + \frac{5 e ^{5 x} cos ( 7 x )}{74}

Add a constant to the solution

= \frac{7 e ^{5 x} sin ( 7 x )}{74} + \frac{5 e ^{5 x} cos ( 7 x )}{74} + C

(3, 8), (7, - 7)

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\int_{1}^{4} 2 π x (e^{x} - ln (x)) d x

(4 arccos (1 - x^{2}))^{^{'}}

\frac{d}{d x} (x^{2} + y^{2} - 9)

What is the integral of e^{5x}cos(7x) ?
The integral of e^{5x}cos(7x) is (7e^{5x}sin(7x))/(74)+(5e^{5x}cos(7x))/(74)+C