What is the integral from-pi to pi/2 of cos(nx) ?

The integral from-pi to pi/2 of cos(nx) is (sin((pin)/2))/(n)

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integral from-pi to pi/2 of cos(nx)

\int_{- π}^{\frac{π}{2}} cos (n x) d x

\frac{sin ( \frac{πn}{2} )}{n}

Solution steps

\int_{- π}^{\frac{π}{2}} cos (n x) d x

Apply u-substitution

: \int_{- πn}^{\frac{πn}{2}} \frac{cos ( u )}{n} d u

= \int_{- πn}^{\frac{πn}{2}} \frac{cos ( u )}{n} d u

Take the constant out:

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

= \frac{1}{n} \cdot \int_{- πn}^{\frac{πn}{2}} cos (u) d u

Use the common integral:

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

= \frac{1}{n} [sin (u)]_{- πn}^{\frac{πn}{2}}

Compute the boundaries:

sin (\frac{πn}{2})

= \frac{1}{n} sin (\frac{πn}{2})

Simplify

= \frac{sin ( \frac{πn}{2} )}{n}

\int_{0}^{0.5} x cos (π x) d x

\int_{2}^{3} x + 3 d x

\int_{4}^{6} (\frac{5}{3} x - 5) d x

\int_{- 1}^{5} 4 x^{3} - 3 x^{2} + 1 d x

\int_{1}^{4} \frac{1}{( x - 3 ) ^{\frac{1}{5}}} d x

What is the integral from-pi to pi/2 of cos(nx) ?
The integral from-pi to pi/2 of cos(nx) is (sin((pin)/2))/(n)