What is the integral of cos^7(xta)n^3x ?

The integral of cos^7(xta)n^3x is

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integral of cos^7(xta)n^3x

\int cos^{7} (x t a) n^{3} x d x

\frac{n ^{3}}{3675 a ^{2} t ^{2}} (3675 a t x sin (a t x) - 3675 a t x sin^{3} (a t x) + 2205 a t x sin^{5} (a t x) - 525 a t x sin^{7} (a t x) + 280 cos^{3} (a t x) + 126 cos^{5} (a t x) + 75 cos^{7} (a t x) + 1680 cos (a t x)) + C

Solution steps

\int cos^{7} (x t a) 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^{7} (a t x) x d x

Apply u-substitution

: \int \frac{u cos ^{7} ( u )}{a ^{2} t ^{2}} d u

= n^{3} \cdot \int \frac{u cos ^{7} ( u )}{a ^{2} t ^{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}{a ^{2} t ^{2}} \cdot \int u cos^{7} (u) d u

Apply Integration By Parts

: u (sin (u) - sin^{3} (u) + \frac{3}{5} sin^{5} (u) - \frac{1}{7} sin^{7} (u)) - \int sin (u) - sin^{3} (u) + \frac{3}{5} sin^{5} (u) - \frac{1}{7} sin^{7} (u) d u

= n^{3} \frac{1}{a ^{2} t ^{2}} (u (sin (u) - sin^{3} (u) + \frac{3}{5} sin^{5} (u) - \frac{1}{7} sin^{7} (u)) - \int sin (u) - sin^{3} (u) + \frac{3}{5} sin^{5} (u) - \frac{1}{7} sin^{7} (u) d u)

\int sin (u) - sin^{3} (u) + \frac{3}{5} sin^{5} (u) - \frac{1}{7} sin^{7} (u) d u = - \frac{16}{35} cos (u) + \frac{9}{35} cos^{3} (u) - \frac{3}{25} cos^{5} (u) - \frac{1}{3} cos^{3} (u) + \frac{3}{35} cos^{5} (u) - \frac{1}{49} cos^{7} (u)

= n^{3} \frac{1}{a ^{2} t ^{2}} (u (sin (u) - sin^{3} (u) + \frac{3}{5} sin^{5} (u) - \frac{1}{7} sin^{7} (u)) - (- \frac{16}{35} cos (u) + \frac{9}{35} cos^{3} (u) - \frac{3}{25} cos^{5} (u) - \frac{1}{3} cos^{3} (u) + \frac{3}{35} cos^{5} (u) - \frac{1}{49} cos^{7} (u)))

Substitute back

u = a t x

= n^{3} \frac{1}{a ^{2} t ^{2}} (a t x (sin (a t x) - sin^{3} (a t x) + \frac{3}{5} sin^{5} (a t x) - \frac{1}{7} sin^{7} (a t x)) - (- \frac{16}{35} cos (a t x) + \frac{9}{35} cos^{3} (a t x) - \frac{3}{25} cos^{5} (a t x) - \frac{1}{3} cos^{3} (a t x) + \frac{3}{35} cos^{5} (a t x) - \frac{1}{49} cos^{7} (a t x)))

Simplify

n^{3} \frac{1}{a ^{2} t ^{2}} (a t x (sin (a t x) - sin^{3} (a t x) + \frac{3}{5} sin^{5} (a t x) - \frac{1}{7} sin^{7} (a t x)) - (- \frac{16}{35} cos (a t x) + \frac{9}{35} cos^{3} (a t x) - \frac{3}{25} cos^{5} (a t x) - \frac{1}{3} cos^{3} (a t x) + \frac{3}{35} cos^{5} (a t x) - \frac{1}{49} cos^{7} (a t x))) : \frac{n ^{3}}{3675 a ^{2} t ^{2}} (3675 a t x sin (a t x) - 3675 a t x sin^{3} (a t x) + 2205 a t x sin^{5} (a t x) - 525 a t x sin^{7} (a t x) + 280 cos^{3} (a t x) + 126 cos^{5} (a t x) + 75 cos^{7} (a t x) + 1680 cos (a t x))

= \frac{n ^{3}}{3675 a ^{2} t ^{2}} (3675 a t x sin (a t x) - 3675 a t x sin^{3} (a t x) + 2205 a t x sin^{5} (a t x) - 525 a t x sin^{7} (a t x) + 280 cos^{3} (a t x) + 126 cos^{5} (a t x) + 75 cos^{7} (a t x) + 1680 cos (a t x))

Add a constant to the solution

= \frac{n ^{3}}{3675 a ^{2} t ^{2}} (3675 a t x sin (a t x) - 3675 a t x sin^{3} (a t x) + 2205 a t x sin^{5} (a t x) - 525 a t x sin^{7} (a t x) + 280 cos^{3} (a t x) + 126 cos^{5} (a t x) + 75 cos^{7} (a t x) + 1680 cos (a t x)) + C

\int - \frac{1}{5 - x} d x

\frac{\partial}{\partial y} (x y)

d P - k P d t = 0

\int 7 sin (2 t) d t

f (x) = x^{2} - 4 x, g (x) = 0, x = 0, x = 4