What is the integral of cos(n)x ?

The integral of cos(n)x is 1/2 x^2cos(n)+C

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integral of cos(n)x

\int cos (n) x d x

\frac{1}{2} x^{2} cos (n) + C

Solution steps

\int cos (n) x d x

Take the constant out:

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

= cos (n) \cdot \int x d x

Apply the Power Rule

: \frac{x ^{2}}{2}

= cos (n) \frac{x ^{2}}{2}

Simplify

cos (n) \frac{x ^{2}}{2} : \frac{1}{2} x^{2} cos (n)

= \frac{1}{2} x^{2} cos (n)

Add a constant to the solution

= \frac{1}{2} x^{2} cos (n) + C

y = 2 x^{3} - x^{2} + \frac{4}{x}, (4, 1)

x y^{^{''}} + y^{^{'}} = 4 x

y^{^{'}} = 2 (1 + y^{2})

x \to 0 - lim (\frac{5}{2 x})

e^{x} + x e^{x}