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

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

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integral of cos(x)e^{ax}

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

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

Solution steps

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

Apply Integration By Parts

: e^{a x} sin (x) - \int e^{a x} a sin (x) d x

= e^{a x} sin (x) - \int e^{a x} a sin (x) d x

Take the constant out:

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

= e^{a x} sin (x) - a \cdot \int e^{a x} sin (x) d x

Apply Integration By Parts

: - e^{a x} cos (x) - \int - a e^{a x} cos (x) d x

= e^{a x} sin (x) - a (- e^{a x} cos (x) - \int - a e^{a x} cos (x) d x)

Take the constant out:

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

= e^{a x} sin (x) - a (- e^{a x} cos (x) - (- a \cdot \int e^{a x} cos (x) d x))

Therefore

\int e^{a x} cos (x) d x = e^{a x} sin (x) - a (- e^{a x} cos (x) - (- a \cdot \int e^{a x} cos (x) d x))

Isolate

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

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

Add a constant to the solution

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

t (ln (2 t))^{2}

3 x (a^{2} - x^{2})

lo g_{8} (x^{2} - 3 x)

\int (ln (x + 1)) d x

\int sin (y) cos (y) d y

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