What is the integral of 10e^{5x}cos(5x) ?

The integral of 10e^{5x}cos(5x) is e^{5x}sin(5x)+e^{5x}cos(5x)+C

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integral of 10e^{5x}cos(5x)

\int 10 e^{5 x} cos (5 x) d x

e^{5 x} sin (5 x) + e^{5 x} cos (5 x) + C

Solution steps

\int 10 e^{5 x} cos (5 x) d x

Take the constant out:

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

= 10 \cdot \int e^{5 x} cos (5 x) d x

Apply Integration By Parts

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

= 10 (\frac{1}{5} e^{5 x} sin (5 x) - \int e^{5 x} sin (5 x) d x)

Apply Integration By Parts

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

= 10 (\frac{1}{5} e^{5 x} sin (5 x) - (- \frac{1}{5} e^{5 x} cos (5 x) - \int - e^{5 x} cos (5 x) d x))

Take the constant out:

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

= 10 (\frac{1}{5} e^{5 x} sin (5 x) - (- \frac{1}{5} e^{5 x} cos (5 x) - (- \int e^{5 x} cos (5 x) d x)))

Therefore

10 \cdot \int e^{5 x} cos (5 x) d x = 10 (\frac{1}{5} e^{5 x} sin (5 x) - (- \frac{1}{5} e^{5 x} cos (5 x) - (- \int e^{5 x} cos (5 x) d x)))

Isolate

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

= 10 (\frac{e ^{5 x} sin ( 5 x )}{10} + \frac{e ^{5 x} cos ( 5 x )}{10})

Simplify

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

= e^{5 x} sin (5 x) + e^{5 x} cos (5 x)

Add a constant to the solution

= e^{5 x} sin (5 x) + e^{5 x} cos (5 x) + C

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What is the integral of 10e^{5x}cos(5x) ?
The integral of 10e^{5x}cos(5x) is e^{5x}sin(5x)+e^{5x}cos(5x)+C