What is the integral from 0 to 1/2 of 4cos(pix) ?

The integral from 0 to 1/2 of 4cos(pix) is 4/pi

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integral from 0 to 1/2 of 4cos(pix)

\int_{0}^{\frac{1}{2}} 4 cos (π x) d x

\frac{4}{π}

Decimal

1.27323 \dots

Solution steps

\int_{0}^{\frac{1}{2}} 4 cos (π x) d x

Take the constant out:

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

= 4 \cdot \int_{0}^{\frac{1}{2}} cos (π x) d x

Apply u-substitution

: \int_{0}^{\frac{π}{2}} cos (u) \frac{1}{π} d u

= 4 \cdot \int_{0}^{\frac{π}{2}} cos (u) \frac{1}{π} d u

Take the constant out:

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

= 4 \cdot \frac{1}{π} \cdot \int_{0}^{\frac{π}{2}} cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= 4 \cdot \frac{1}{π} [sin (u)]_{0}^{\frac{π}{2}}

Simplify

4 \cdot \frac{1}{π} [sin (u)]_{0}^{\frac{π}{2}} : \frac{4}{π} [sin (u)]_{0}^{\frac{π}{2}}

= \frac{4}{π} [sin (u)]_{0}^{\frac{π}{2}}

Compute the boundaries:

1

= \frac{4}{π} \cdot 1

Simplify

= \frac{4}{π}

\int x \cdot \frac{1}{x ^{2}} d x

x \to - 3 lim (- 2 x^{3})

f (x) = x + 2, at x = - 1

x \to 1 + lim (x^{2} - 3)

\frac{d}{d x} (- 6 e^{5 x})

What is the integral from 0 to 1/2 of 4cos(pix) ?
The integral from 0 to 1/2 of 4cos(pix) is 4/pi