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

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

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

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

\frac{n ^{2}}{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^{2} x d x

Take the constant out:

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

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

Apply u-substitution

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

Apply Integration By Parts

= n^{2} \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^{2} \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^{2} \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^{2} \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 ^{2}}{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 ^{2}}{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 ^{2}}{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

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