What is the integral of (8x)/(sqrt(81-16x^4)) ?

The integral of (8x)/(sqrt(81-16x^4)) is arcsin(4/9 x^2)+C

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integral of (8x)/(sqrt(81-16x^4))

\int \frac{8 x}{81 - 16 x ^{4}} d x

arcsin (\frac{4}{9} x^{2}) + C

Solution steps

\int \frac{8 x}{81 - 16 x ^{4}} d x

Take the constant out:

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

= 8 \cdot \int \frac{x}{81 - 16 x ^{4}} d x

Apply u-substitution

: \int \frac{1}{2 81 - 16 u ^{2}} d u

= 8 \cdot \int \frac{1}{2 81 - 16 u ^{2}} d u

Take the constant out:

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

= 8 \cdot \frac{1}{2} \cdot \int \frac{1}{81 - 16 u ^{2}} d u

Apply Trigonometric Substitution

: \int \frac{1}{4} d v

= 8 \cdot \frac{1}{2} \cdot \int \frac{1}{4} d v

Take the constant out:

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

= 8 \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 8 \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot 1 \cdot v

Substitute back

= 8 \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot 1 \cdot arcsin (\frac{4}{9} x^{2})

Simplify

8 \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot 1 \cdot arcsin (\frac{4}{9} x^{2}) : arcsin (\frac{4}{9} x^{2})

= arcsin (\frac{4}{9} x^{2})

Add a constant to the solution

= arcsin (\frac{4}{9} x^{2}) + C

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What is the integral of (8x)/(sqrt(81-16x^4)) ?
The integral of (8x)/(sqrt(81-16x^4)) is arcsin(4/9 x^2)+C