integral from 0 to pi/4 of 4sin^2(x)

1) Let's compute the integral:

\int_{0}^{\frac{π}{4}} 4 \cdot {\sin (x)}^{2} d x

We reduce the square using the identity

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

ⅰ

\int 4 \cdot {\sin (x)}^{2} d x

\int (2 - 2 \cdot \cos (2 x)) d x

\int (2 - 2 \cdot \cos (2 x)) d x

For the math expression:

- 2 \cdot \cos (2 x)

, since (sin(a))' = a'*cos(a), we try differentiating

\sin (2 x)

2) Let's perform the differentiation:

\frac{d}{d x} (\sin (2 x))

We differentiate using:

\frac{d}{d x} (\sin (f)) = \frac{d}{d x} (f) \cdot \cos (f)

(where f =

2 x

)

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (2 x))

\frac{d}{d x} (2 x) \cdot \cos (2 x)

\frac{d}{d x} (2 x) \cdot \cos (2 x)

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (2 x)

2

2 \cdot \cos (2 x)

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⅰ

\int (2 - 2 \cdot \cos (2 x)) d x

- \sin (2 x) + \int 2 d x

- \sin (2 x) + \int 2 d x

ⓘ The derivative of

2 x

2

, so

2 x

is an antiderivative of

2

\int 2 d x

2 x

▷

- \sin (2 x) + 2 x

3) Let's evaluate for the interval

[0, \frac{π}{4}]

(2 \cdot \frac{π}{4} - \sin (2 \cdot \frac{π}{4})) - (2 \cdot 0 - \sin (2 \cdot 0))

ⓘ Multiplying a fraction by a number multiplies its numerator:

2 \cdot \frac{π}{4}

\frac{2 π}{4}

■A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{2 π}{4}

\frac{π}{2}

■A minus sign before a parenthesis changes the sign of each of its terms:

- (2 \cdot 0 - \sin (2 \cdot 0))

- 2 \cdot 0 + \sin (2 \cdot 0)

\frac{π}{2}

- \sin (\frac{π}{2})

- 2 \cdot 0

+ \sin (2 \cdot 0)

ⓘ The sine is the y-coordinate (vertical axis) of the point reached on the unit circle after a rotation of

\frac{π}{2}

radians from the horizontal axis pointing to the right, this is a remarkable value:

- \sin (\frac{π}{2})

-1

■A product with a zero factor is zero:

-2 \cdot 0

0

2 \cdot 0

0

\frac{π}{2}

- 1

+ 0

+ \sin (0)

ⓘ Adding 0 changes nothing:

0 + \sin (0)

\sin (0)

■The sine is the y-coordinate (vertical axis) of the point reached on the unit circle after a rotation of

0

radians from the horizontal axis pointing to the right, this is a remarkable value:

\sin (0)

0

\frac{π}{2} - 1 + 0

ⓘ Adding 0 changes nothing:

- 1 + 0

-1

▶Integral:

\frac{π}{2} - 1

4) The numeric value of:

\frac{π}{2} - 1

is:

\frac{\approx 3.1416}{2} - 1

\approx 1.5708 - 1

▷

\approx 0.5708