What is the integral of x^{11}cos(x^6) ?

The integral of x^{11}cos(x^6) is 1/6 (x^6sin(x^6)+cos(x^6))+C

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integral of x^{11}cos(x^6)

\int x^{11} cos (x^{6}) d x

\frac{1}{6} (x^{6} sin (x^{6}) + cos (x^{6})) + C

Solution steps

\int x^{11} cos (x^{6}) d x

Apply u-substitution

: \int \frac{u cos ( u )}{6} d u

= \int \frac{u cos ( u )}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{1}{6} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= \frac{1}{6} (u sin (u) - (- cos (u)))

Substitute back

u = x^{6}

= \frac{1}{6} (x^{6} sin (x^{6}) - (- cos (x^{6})))

Simplify

= \frac{1}{6} (x^{6} sin (x^{6}) + cos (x^{6}))

Add a constant to the solution

= \frac{1}{6} (x^{6} sin (x^{6}) + cos (x^{6})) + C

x \to 0 lim (x^{4} cos (\frac{2}{x}))

\int csc^{3} (x) d x

\int \frac{3 e ^{3 x}}{7 + e ^{3 x}} d x

\frac{d}{d x} (x^{3} - x^{2} - 6 x + 2)

What is the integral of x^{11}cos(x^6) ?
The integral of x^{11}cos(x^6) is 1/6 (x^6sin(x^6)+cos(x^6))+C