What is the integral of (4x^3)/(sqrt(1-x^8)) ?

The integral of (4x^3)/(sqrt(1-x^8)) is arcsin(x^4)+C

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integral of (4x^3)/(sqrt(1-x^8))

\int \frac{4 x ^{3}}{1 - x ^{8}} d x

arcsin (x^{4}) + C

Solution steps

\int \frac{4 x ^{3}}{1 - x ^{8}} d x

Take the constant out:

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

= 4 \cdot \int \frac{x ^{3}}{1 - x ^{8}} d x

Apply u-substitution

: \int \frac{1}{4 1 - u ^{2}} d u

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

Take the constant out:

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

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

Use the common integral:

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

= 4 \cdot \frac{1}{4} arcsin (u)

Substitute back

u = x^{4}

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

Simplify

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

= arcsin (x^{4})

Add a constant to the solution

= arcsin (x^{4}) + C

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What is the integral of (4x^3)/(sqrt(1-x^8)) ?
The integral of (4x^3)/(sqrt(1-x^8)) is arcsin(x^4)+C