What is the integral of xsqrt(36-x^4) ?

The integral of xsqrt(36-x^4) is 9(arcsin(1/6 x^2)+1/2 sin(2arcsin(1/6 x^2)))+C

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integral of xsqrt(36-x^4)

\int x 36 - x^{4} d x

9 (arcsin (\frac{1}{6} x^{2}) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} x^{2}))) + C

Solution steps

\int x 36 - x^{4} d x

Apply u-substitution

: \int \frac{36 - u}{4 u} d u

= \int \frac{36 - u}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{36 - u}{u} d u

Apply u-substitution

: \int 2 36 - v^{2} d v

= \frac{1}{4} \cdot \int 2 36 - v^{2} d v

Take the constant out:

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

= \frac{1}{4} \cdot 2 \cdot \int 36 - v^{2} d v

Apply Trigonometric Substitution

: \int 36 cos^{2} (w) d w

= \frac{1}{4} \cdot 2 \cdot \int 36 cos^{2} (w) d w

Take the constant out:

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

= \frac{1}{4} \cdot 2 \cdot 36 \cdot \int cos^{2} (w) d w

Rewrite using trig identities

= \frac{1}{4} \cdot 2 \cdot 36 \cdot \int \frac{1 + cos ( 2 w )}{2} d w

Take the constant out:

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

= \frac{1}{4} \cdot 2 \cdot 36 \cdot \frac{1}{2} \cdot \int 1 + cos (2 w) d w

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{4} \cdot 2 \cdot 36 \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= \frac{1}{4} \cdot 2 \cdot 36 \cdot \frac{1}{2} (w + \frac{1}{2} sin (2 w))

Substitute back

= \frac{1}{4} \cdot 2 \cdot 36 \cdot \frac{1}{2} (arcsin (\frac{1}{6} x^{4}) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} x^{4})))

Simplify

\frac{1}{4} \cdot 2 \cdot 36 \cdot \frac{1}{2} (arcsin (\frac{1}{6} x^{4}) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} x^{4}))) : 9 (arcsin (\frac{1}{6} x^{2}) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} x^{2})))

= 9 (arcsin (\frac{1}{6} x^{2}) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} x^{2})))

Add a constant to the solution

= 9 (arcsin (\frac{1}{6} x^{2}) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} x^{2}))) + C

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What is the integral of xsqrt(36-x^4) ?
The integral of xsqrt(36-x^4) is 9(arcsin(1/6 x^2)+1/2 sin(2arcsin(1/6 x^2)))+C