What is the integral of sin^2((pix)/4) ?

The integral of sin^2((pix)/4) is 1/2 x-1/pi sin((pix)/2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin^2((pix)/4)

\int sin^{2} (\frac{π x}{4}) d x

\frac{1}{2} x - \frac{1}{π} sin (\frac{π x}{2}) + C

Solution steps

\int sin^{2} (\frac{π x}{4}) d x

Rewrite using trig identities

= \int \frac{1}{2} (1 - cos (\frac{π x}{2})) d x

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 (\frac{π x}{2}) d x

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 x - \int cos (\frac{π x}{2}) d x)

\int 1 d x = x

\int cos (\frac{π x}{2}) d x = \frac{2}{π} sin (\frac{π x}{2})

= \frac{1}{2} (x - \frac{2}{π} sin (\frac{π x}{2}))

Simplify

\frac{1}{2} (x - \frac{2}{π} sin (\frac{π x}{2})) : \frac{1}{2} x - \frac{1}{π} sin (\frac{π x}{2})

= \frac{1}{2} x - \frac{1}{π} sin (\frac{π x}{2})

Add a constant to the solution

= \frac{1}{2} x - \frac{1}{π} sin (\frac{π x}{2}) + C

\int \frac{1}{3 ^{2} + x ^{2}} d x

\int \frac{3 x + 2}{x ( x + 2 ) ^{2} + 16 x} d x

\int \frac{3}{[ x ( x + 3 ) ]} d x

\int \frac{( x ^{3} - 4 )}{( x ^{3} + 2 x ^{2} + 2 x )} d x

\int sin^{2} (5 x) \cdot cos^{3} (5 x) d x

What is the integral of sin^2((pix)/4) ?
The integral of sin^2((pix)/4) is 1/2 x-1/pi sin((pix)/2)+C