What is the integral of sec^{-2}(xta)n^3x ?

The integral of sec^{-2}(xta)n^3x is (n^3)/(8a^2t^2)(2a^2t^2x^2+2atxsin(2atx)+cos(2atx))+C

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integral of sec^{-2}(xta)n^3x

\int sec^{- 2} (x t a) n^{3} x d x

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

Solution steps

\int sec^{- 2} (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 sec^{- 2} (a t x) x d x

Simplify

sec^{- 2} (a t x) x : x cos^{2} (a t x)

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

Apply u-substitution

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

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

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

Simplify

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

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

Add a constant to the solution

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

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

\frac{d}{d x} (4 x^{2} + x + 1)

What is the integral of sec^{-2}(xta)n^3x ?
The integral of sec^{-2}(xta)n^3x is (n^3)/(8a^2t^2)(2a^2t^2x^2+2atxsin(2atx)+cos(2atx))+C