What is the integral of x^3sqrt(a^2x^2-b^2) ?

The integral of x^3sqrt(a^2x^2-b^2) is (5b^2(a^2x^2-b^2)^{3/2}+3(a^2x^2-b^2)^{5/2})/(15a^4)+C

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integral of x^3sqrt(a^2x^2-b^2)

\int x^{3} a^{2} x^{2} - b^{2} d x

\frac{5 b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{3}{2}} + 3 ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{5}{2}}}{15 a ^{4}} + C

Solution steps

\int x^{3} a^{2} x^{2} - b^{2} d x

Apply Trigonometric Substitution

: \int \frac{b ^{5} sec ^{4} ( u ) tan ^{2} ( u )}{a ^{4}} d u

= \int \frac{b ^{5} sec ^{4} ( u ) tan ^{2} ( u )}{a ^{4}} d u

Take the constant out:

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

= \frac{b ^{5}}{a ^{4}} \cdot \int sec^{4} (u) tan^{2} (u) d u

Rewrite using trig identities

= \frac{b ^{5}}{a ^{4}} \cdot \int (1 + tan^{2} (u)) sec^{2} (u) tan^{2} (u) d u

Apply u-substitution

: \int v^{2} (1 + v^{2}) d v

= \frac{b ^{5}}{a ^{4}} \cdot \int v^{2} (1 + v^{2}) d v

Expand

v^{2} (1 + v^{2}) : v^{2} + v^{4}

= \frac{b ^{5}}{a ^{4}} \cdot \int v^{2} + v^{4} d v

Apply the Sum Rule:

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

= \frac{b ^{5}}{a ^{4}} (\int v^{2} d v + \int v^{4} d v)

\int v^{2} d v = \frac{v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= \frac{b ^{5}}{a ^{4}} (\frac{v ^{3}}{3} + \frac{v ^{5}}{5})

Substitute back

= \frac{b ^{5}}{a ^{4}} (\frac{tan ^{3} ( arcsec ( \frac{a}{b} x ) )}{3} + \frac{tan ^{5} ( arcsec ( \frac{a}{b} x ) )}{5})

Simplify

\frac{b ^{5}}{a ^{4}} (\frac{tan ^{3} ( arcsec ( \frac{a}{b} x ) )}{3} + \frac{tan ^{5} ( arcsec ( \frac{a}{b} x ) )}{5}) : \frac{5 b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{3}{2}} + 3 ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{5}{2}}}{15 a ^{4}}

= \frac{5 b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{3}{2}} + 3 ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{5}{2}}}{15 a ^{4}}

Add a constant to the solution

= \frac{5 b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{3}{2}} + 3 ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{5}{2}}}{15 a ^{4}} + C

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What is the integral of x^3sqrt(a^2x^2-b^2) ?
The integral of x^3sqrt(a^2x^2-b^2) is (5b^2(a^2x^2-b^2)^{3/2}+3(a^2x^2-b^2)^{5/2})/(15a^4)+C