What is the integral of x^2cos(xy) ?

The integral of x^2cos(xy) is xsin(xy)+C

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integral of x^2cos(xy)

\int x^{2} cos (x y) d y

x sin (x y) + C

Solution steps

\int x^{2} cos (x y) d y

Take the constant out:

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

= x^{2} \cdot \int cos (x y) d y

Apply u-substitution

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

= x^{2} \cdot \int \frac{cos ( u )}{x} d u

Take the constant out:

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

= x^{2} \frac{1}{x} \cdot \int cos (u) d u

Use the common integral:

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

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

Substitute back

u = x y

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

Simplify

x^{2} \frac{1}{x} sin (x y) : x sin (x y)

= x sin (x y)

Add a constant to the solution

= x sin (x y) + C

f (t) = e^{4 t}

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

t \to 0 lim (\frac{6 e ^{t} - 6}{t})

\int - e^{x} cos (x) d x

\int \frac{1 + x}{2 x} d x