integral of -2sin(x)^2 from 0 to pi — step-by-step solution

1) Let's compute the integral:

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

We reduce the square using the identity

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

ⅰ

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

- \int (1 - \cos (2 x)) d x

- \int (1 - \cos (2 x)) d x

For the math expression:

- \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))

Apply the differentiation rule:

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

with 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)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⅰ

- \int (1 - \cos (2 x)) d x

- (- \frac{1}{2} \cdot \sin (2 x) + \int 1 d x)

- (- \frac{1}{2} \cdot \sin (2 x) + \int 1 d x)

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{2} \cdot \sin (2 x)

\frac{1 \cdot \sin (2 x)}{2}

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

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

- (- \frac{\sin (2 x)}{2} + \int 1 d x)

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

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

ⓘ The derivative of

- x

-1

, so

- x

is an antiderivative of

-1

- \int 1 d x

- x

▷

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

3) Let's evaluate over the interval

[0, π]

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

ⓘ cos and sin are periodic with period 2π:

\sin (2 π)

\sin (0)

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

- (\frac{\sin (2 \cdot 0)}{2} + 0)

- \frac{\sin (2 \cdot 0)}{2} + 0

\frac{\sin (0)}{2}

- π

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

+ 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

■A product with a zero factor is zero:

2 \cdot 0

0

■Adding 0 changes nothing:

- \frac{\sin (0)}{2} + 0

- \frac{\sin (0)}{2}

\frac{0}{2} - π - \frac{\sin (0)}{2}

ⓘ 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

0 - π - \frac{0}{2}

ⓘ Adding 0 changes nothing:

0 - π

- π

- π + 0

ⓘ Adding 0 changes nothing:

- π + 0

- π

▶Integral:

- π

4) The numeric value of:

- π

is:

▷

\approx -3.1416