What is the integral of x^5cos(x^2) ?

The integral of x^5cos(x^2) is 1/2 x^4sin(x^2)+x^2cos(x^2)-sin(x^2)+C

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integral of x^5cos(x^2)

\int x^{5} cos (x^{2}) d x

\frac{1}{2} x^{4} sin (x^{2}) + x^{2} cos (x^{2}) - sin (x^{2}) + C

Solution steps

\int x^{5} cos (x^{2}) d x

Apply u-substitution

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

= \int \frac{u ^{2} cos ( u )}{2} d u

Take the constant out:

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

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

Apply Integration By Parts

: u^{2} sin (u) - \int 2 u sin (u) d u

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

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

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

Substitute back

u = x^{2}

= \frac{1}{2} ((x^{2})^{2} sin (x^{2}) - 2 (- x^{2} cos (x^{2}) + sin (x^{2})))

Simplify

\frac{1}{2} ((x^{2})^{2} sin (x^{2}) - 2 (- x^{2} cos (x^{2}) + sin (x^{2}))) : \frac{1}{2} x^{4} sin (x^{2}) + x^{2} cos (x^{2}) - sin (x^{2})

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

Add a constant to the solution

= \frac{1}{2} x^{4} sin (x^{2}) + x^{2} cos (x^{2}) - sin (x^{2}) + C

x \to 1 - lim (\frac{7}{1 - x})

\int \frac{x}{1 - 4 x ^{2}} d x

\int \frac{5 x}{( 7 - x ^{2} ) ^{3}} d x

n = 1 \sum \infty \frac{1}{n ^{1.5}}

x \to 0 lim (\frac{3 x ^{4} - 8 x}{x})

What is the integral of x^5cos(x^2) ?
The integral of x^5cos(x^2) is 1/2 x^4sin(x^2)+x^2cos(x^2)-sin(x^2)+C