What is the integral of e^xsin(x) ?

The integral of e^xsin(x) is -(e^xcos(x))/2+(e^xsin(x))/2+C

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integral of e^xsin(x)

\int e^{x} sin (x) d x

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

Solution steps

\int e^{x} sin (x) d x

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

\int e^{x} sin (x) d x

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

Add a constant to the solution

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

f (x) = \frac{1}{27} (9 x^{2} + 6)^{\frac{3}{2}}

\int e^{\frac{1}{2} x^{2}} d x

x \to 4 - lim (∣ x - 4 ∣)

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

\frac{\partial}{\partial x} (3 x y^{2} + 2 y cos (x))

What is the integral of e^xsin(x) ?
The integral of e^xsin(x) is -(e^xcos(x))/2+(e^xsin(x))/2+C