What is the integral of e^{x/4}sin(x/8)cos(x/8) ?

The integral of e^{x/4}sin(x/8)cos(x/8) is e^{x/4}sin(x/4)-e^{x/4}cos(x/4)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^{x/4}sin(x/8)cos(x/8)

\int e^{\frac{x}{4}} sin (\frac{x}{8}) cos (\frac{x}{8}) d x

e^{\frac{x}{4}} sin (\frac{x}{4}) - e^{\frac{x}{4}} cos (\frac{x}{4}) + C

Solution steps

\int e^{\frac{x}{4}} sin (\frac{x}{8}) cos (\frac{x}{8}) d x

Apply u-substitution

: \int 8 e^{2 u} cos (u) sin (u) d u

= \int 8 e^{2 u} cos (u) sin (u) d u

Take the constant out:

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

= 8 \cdot \int e^{2 u} cos (u) sin (u) d u

Rewrite using trig identities

= 8 \cdot \int \frac{1}{2} e^{2 u} sin (2 u) d u

Take the constant out:

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

= 8 \cdot \frac{1}{2} \cdot \int e^{2 u} sin (2 u) d u

Apply Integration By Parts

: - \frac{1}{2} e^{2 u} cos (2 u) - \int - e^{2 u} cos (2 u) d u

= 8 \cdot \frac{1}{2} (- \frac{1}{2} e^{2 u} cos (2 u) - \int - e^{2 u} cos (2 u) d u)

Take the constant out:

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

= 8 \cdot \frac{1}{2} (- \frac{1}{2} e^{2 u} cos (2 u) - (- \int e^{2 u} cos (2 u) d u))

Apply Integration By Parts

: \frac{1}{2} e^{2 u} sin (2 u) - \int e^{2 u} sin (2 u) d u

= 8 \cdot \frac{1}{2} (- \frac{1}{2} e^{2 u} cos (2 u) - (- (\frac{1}{2} e^{2 u} sin (2 u) - \int e^{2 u} sin (2 u) d u)))

Therefore

8 \cdot \frac{1}{2} \cdot \int e^{2 u} sin (2 u) d u = 8 \cdot \frac{1}{2} (- \frac{1}{2} e^{2 u} cos (2 u) - (- (\frac{1}{2} e^{2 u} sin (2 u) - \int e^{2 u} sin (2 u) d u)))

Isolate

\int e^{2 u} sin (2 u) d u

= 8 \cdot \frac{1}{2} (\frac{e ^{2 u} sin ( 2 u )}{4} - \frac{e ^{2 u} cos ( 2 u )}{4})

Substitute back

u = \frac{x}{8}

= 8 \cdot \frac{1}{2} (\frac{e ^{2 \cdot \frac{x}{8}} sin ( 2 \cdot \frac{x}{8} )}{4} - \frac{e ^{2 \cdot \frac{x}{8}} cos ( 2 \cdot \frac{x}{8} )}{4})

Simplify

8 \cdot \frac{1}{2} (\frac{e ^{2 \cdot \frac{x}{8}} sin ( 2 \cdot \frac{x}{8} )}{4} - \frac{e ^{2 \cdot \frac{x}{8}} cos ( 2 \cdot \frac{x}{8} )}{4}) : e^{\frac{x}{4}} sin (\frac{x}{4}) - e^{\frac{x}{4}} cos (\frac{x}{4})

= e^{\frac{x}{4}} sin (\frac{x}{4}) - e^{\frac{x}{4}} cos (\frac{x}{4})

Add a constant to the solution

= e^{\frac{x}{4}} sin (\frac{x}{4}) - e^{\frac{x}{4}} cos (\frac{x}{4}) + C

\int (1 + 2 x)^{5} d x

\int 3 x^{2} x^{3} - 1 d x

\int \frac{x}{( 5 x ^{2} - 4 ) ^{4}} d x

\int \frac{sin ^{5} ( x ) - cos ^{2} ( x )}{cos ( x )} d x

\int \frac{4 x - 12}{2} d x

What is the integral of e^{x/4}sin(x/8)cos(x/8) ?
The integral of e^{x/4}sin(x/8)cos(x/8) is e^{x/4}sin(x/4)-e^{x/4}cos(x/4)+C