What is the integral of sin(xt) ?

The integral of sin(xt) is -1/x cos(xt)+C

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integral of sin(xt)

\int sin (x t) d t

- \frac{1}{x} cos (x t) + C

Solution steps

\int sin (x t) d t

Apply u-substitution

: \int \frac{sin ( u )}{x} d u

= \int \frac{sin ( u )}{x} d u

Take the constant out:

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

= \frac{1}{x} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{x} (- cos (u))

Substitute back

u = x t

= \frac{1}{x} (- cos (x t))

Simplify

= - \frac{1}{x} cos (x t)

Add a constant to the solution

= - \frac{1}{x} cos (x t) + C

\frac{d}{d x} (x + y + z)

\int \frac{2 x ^{3} + 6 x ^{2} + 7 x + 4}{x ^{2} + 3 x + 2} d x

y = \frac{1}{x}, (1, 1)

\frac{d}{d x} ((x^{2} + 3 x - 9) e^{x})

\frac{d}{d x} (x^{3} + k y (x)^{2} - 5 x y)