What is expand (ux^3-ln(x))(xdu+udx) ?

The solution to expand (ux^3-ln(x))(xdu+udx) is ux^4du+u^2x^3dx-xln(x)du-uln(x)dx

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expand (ux^3-ln(x))(xdu+udx)

expand

(u x^{3} - ln (x)) (x d u + u d x)

u x^{4} d u + u^{2} x^{3} d x - x ln (x) d u - u ln (x) d x

Solution steps

(u x^{3} - ln (x)) (x d u + u d x)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = u x^{3}, b = - ln (x), c = x d u, d = u d x

= u x^{3} x d u + u x^{3} u d x + (- ln (x)) x d u + (- ln (x)) u d x

Apply minus-plus rules

+ (- a) = - a

= u x^{3} x d u + uu x^{3} d x - x ln (x) d u - u ln (x) d x

Simplify

u x^{3} x d u + uu x^{3} d x - x ln (x) d u - u ln (x) d x : u x^{4} d u + u^{2} x^{3} d x - x ln (x) d u - u ln (x) d x

= u x^{4} d u + u^{2} x^{3} d x - x ln (x) d u - u ln (x) d x

\frac{( x + 1 ) ( x + 4 )}{( x - 3 ) ( x + 5 ) ( x - 7 )}

([(1 + 0.22) - 1]^{1})^{2}

\frac{( a ^{2} + b ^{2} ) ( b - 2 a ) ^{2}}{b ^{2} ( 4 a ^{2} - 4 ab + b ) ^{2}}

\frac{1}{( s + 1 ) \cdot ( s + 2 ) ^{2}}

\frac{\partial}{\partial y} ((z) (x - z - arctan (yz)))

What is expand (ux^3-ln(x))(xdu+udx) ?
The solution to expand (ux^3-ln(x))(xdu+udx) is ux^4du+u^2x^3dx-xln(x)du-uln(x)dx