What is the integral of e^x(x+sin(e^x)) ?

The integral of e^x(x+sin(e^x)) is e^xx-e^x-cos(e^x)+C

Popular >

integral of e^x(x+sin(e^x))

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

e^{x} x - e^{x} - cos (e^{x}) + C

Solution steps

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

Apply u-substitution

: \int ln (u) + sin (u) d u

= \int ln (u) + sin (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int ln (u) d u + \int sin (u) d u

\int ln (u) d u = u ln (u) - u

\int sin (u) d u = - cos (u)

= u ln (u) - u - cos (u)

Substitute back

u = e^{x}

= e^{x} ln (e^{x}) - e^{x} - cos (e^{x})

Simplify

ln (e^{x}) : x

= e^{x} x - e^{x} - cos (e^{x})

Add a constant to the solution

= e^{x} x - e^{x} - cos (e^{x}) + C

\int (16 - x^{2})^{\frac{1}{2}} d x

\int - \frac{5}{8 t 64 t ^{2} - 81} d t

\int \frac{2 x ^{5} - x + 3}{x} d x

\int 2 x^{3} sin (x) d x

\int (2 x - 1) (8 x^{2} - 8 x + 1)^{\frac{1}{3}} d x

What is the integral of e^x(x+sin(e^x)) ?
The integral of e^x(x+sin(e^x)) is e^xx-e^x-cos(e^x)+C