What is the integral from 0 to pi/3 of xsin(x+y) ?

The integral from 0 to pi/3 of xsin(x+y) is x(-cos(x+pi/3)+cos(x))

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integral from 0 to pi/3 of xsin(x+y)

\int_{0}^{\frac{π}{3}} x sin (x + y) d y

x (- cos (x + \frac{π}{3}) + cos (x))

Solution steps

\int_{0}^{\frac{π}{3}} x sin (x + y) d y

Take the constant out:

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

= x \cdot \int_{0}^{\frac{π}{3}} sin (x + y) d y

Apply u-substitution

: \int_{x}^{x + \frac{π}{3}} sin (u) d u

= x \cdot \int_{x}^{x + \frac{π}{3}} sin (u) d u

Use the common integral:

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

= x [- cos (u)]_{x}^{x + \frac{π}{3}}

Compute the boundaries:

- cos (x + \frac{π}{3}) + cos (x)

= x (- cos (x + \frac{π}{3}) + cos (x))

\int_{x}^{2} \frac{x y}{2} d y

\int_{- 2}^{3} (2 x - 1) d x

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

\int_{2}^{4} (25 - x^{2})^{2} d x

\int_{2}^{3} t 1 + 4 t^{2} d t

What is the integral from 0 to pi/3 of xsin(x+y) ?
The integral from 0 to pi/3 of xsin(x+y) is x(-cos(x+pi/3)+cos(x))