What is the integral of cos^2(xsi)n^3x ?

The integral of cos^2(xsi)n^3x is (n^3)/(8s^2i^2)(2s^2i^2x^2+2sixsin(2six)+cos(2six))+C

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integral of cos^2(xsi)n^3x

\int cos^{2} (x s i) n^{3} x d x

\frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x)) + C

Solution steps

\int cos^{2} (x s i) n^{3} x d x

Take the constant out:

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

= n^{3} \cdot \int cos^{2} (s i x) x d x

Apply u-substitution

: \int \frac{u cos ^{2} ( u )}{s ^{2} i ^{2}} d u

= n^{3} \cdot \int \frac{u cos ^{2} ( u )}{s ^{2} i ^{2}} d u

Take the constant out:

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

= n^{3} \frac{1}{s ^{2} i ^{2}} \cdot \int u cos^{2} (u) d u

Apply Integration By Parts

: \frac{1}{2} u (u + \frac{1}{2} sin (2 u)) - \int \frac{1}{2} (u + \frac{1}{2} sin (2 u)) d u

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} u (u + \frac{1}{2} sin (2 u)) - \int \frac{1}{2} (u + \frac{1}{2} sin (2 u)) d u)

\int \frac{1}{2} (u + \frac{1}{2} sin (2 u)) d u = \frac{1}{2} (\frac{u ^{2}}{2} - \frac{1}{4} cos (2 u))

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} u (u + \frac{1}{2} sin (2 u)) - \frac{1}{2} (\frac{u ^{2}}{2} - \frac{1}{4} cos (2 u)))

Substitute back

u = s i x

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} s i x (s i x + \frac{1}{2} sin (2 s i x)) - \frac{1}{2} (\frac{( s i x ) ^{2}}{2} - \frac{1}{4} cos (2 s i x)))

Simplify

n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} s i x (s i x + \frac{1}{2} sin (2 s i x)) - \frac{1}{2} (\frac{( s i x ) ^{2}}{2} - \frac{1}{4} cos (2 s i x))) : \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x))

= \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x))

Add a constant to the solution

= \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x)) + C

e^{- 2 t}, at t = 0

\frac{d}{d x} (\frac{x}{x - 2 x})

x \to - 1 lim (2 x^{2} + 6 x - 1)

x \to 2 lim (\frac{5}{∣ 2 - x ∣})

\frac{d}{d x} (e^{- \frac{x}{a}})

What is the integral of cos^2(xsi)n^3x ?
The integral of cos^2(xsi)n^3x is (n^3)/(8s^2i^2)(2s^2i^2x^2+2sixsin(2six)+cos(2six))+C