What is the integral of e^{7x}cos(3x) ?

The integral of e^{7x}cos(3x) is (3e^{7x}sin(3x))/(58)+(7e^{7x}cos(3x))/(58)+C

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integral of e^{7x}cos(3x)

\int e^{7 x} cos (3 x) d x

\frac{3 e ^{7 x} sin ( 3 x )}{58} + \frac{7 e ^{7 x} cos ( 3 x )}{58} + C

Solution steps

\int e^{7 x} cos (3 x) d x

Apply Integration By Parts

: \frac{1}{3} e^{7 x} sin (3 x) - \int \frac{7}{3} e^{7 x} sin (3 x) d x

= \frac{1}{3} e^{7 x} sin (3 x) - \int \frac{7}{3} e^{7 x} sin (3 x) d x

Take the constant out:

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

= \frac{1}{3} e^{7 x} sin (3 x) - \frac{7}{3} \cdot \int e^{7 x} sin (3 x) d x

Apply Integration By Parts

: - \frac{1}{3} e^{7 x} cos (3 x) - \int - \frac{7}{3} e^{7 x} cos (3 x) d x

= \frac{1}{3} e^{7 x} sin (3 x) - \frac{7}{3} (- \frac{1}{3} e^{7 x} cos (3 x) - \int - \frac{7}{3} e^{7 x} cos (3 x) d x)

Take the constant out:

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

= \frac{1}{3} e^{7 x} sin (3 x) - \frac{7}{3} (- \frac{1}{3} e^{7 x} cos (3 x) - (- \frac{7}{3} \cdot \int e^{7 x} cos (3 x) d x))

Therefore

\int e^{7 x} cos (3 x) d x = \frac{1}{3} e^{7 x} sin (3 x) - \frac{7}{3} (- \frac{1}{3} e^{7 x} cos (3 x) - (- \frac{7}{3} \cdot \int e^{7 x} cos (3 x) d x))

Isolate

\int e^{7 x} cos (3 x) d x

= \frac{3 e ^{7 x} sin ( 3 x )}{58} + \frac{7 e ^{7 x} cos ( 3 x )}{58}

Add a constant to the solution

= \frac{3 e ^{7 x} sin ( 3 x )}{58} + \frac{7 e ^{7 x} cos ( 3 x )}{58} + C

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

\frac{1}{4} y^{^{''}} + y^{^{'}} + y = x^{2} - 4 x

y \cdot y^{^{'}} + 36 x = 0

y = x, y = \frac{x}{2}

\frac{d}{d x} (a e^{- 2 x})

What is the integral of e^{7x}cos(3x) ?
The integral of e^{7x}cos(3x) is (3e^{7x}sin(3x))/(58)+(7e^{7x}cos(3x))/(58)+C