What is the integral of e^x*cos^2(e^x) ?

The integral of e^x*cos^2(e^x) is 1/2 (e^x+1/2 sin(2e^x))+C

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integral of e^x*cos^2(e^x)

integral

e^{x} \cdot cos^{2} (e^{x})

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

Solution steps

\int e^{x} cos^{2} (e^{x}) d x

Apply u-substitution

= \int cos^{2} (u) d u

Rewrite using trig identities

= \int \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 + cos (2 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

= \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = e^{x}

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

Add a constant to the solution

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

What is the integral of e^x*cos^2(e^x) ?
The integral of e^x*cos^2(e^x) is 1/2 (e^x+1/2 sin(2e^x))+C