What is the integral of (3x^{14}+x^4)(x^{15}+x^5) ?

The integral of (3x^{14}+x^4)(x^{15}+x^5) is 1/10 x^{10}(x^{20}+2x^{10}+1)+C

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integral of (3x^{14}+x^4)(x^{15}+x^5)

\int (3 x^{14} + x^{4}) (x^{15} + x^{5}) d x

\frac{1}{10} x^{10} (x^{20} + 2 x^{10} + 1) + C

Solution steps

\int (3 x^{14} + x^{4}) (x^{15} + x^{5}) d x

Apply u-substitution

: \int \frac{( u ^{\frac{2}{7}} + 3 u ) ( u ^{\frac{5}{7}} + 1 )}{14 u ^{\frac{4}{7}}} d u

= \int \frac{( u ^{\frac{2}{7}} + 3 u ) ( u ^{\frac{5}{7}} + 1 )}{14 u ^{\frac{4}{7}}} d u

Take the constant out:

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

= \frac{1}{14} \cdot \int \frac{( u ^{\frac{2}{7}} + 3 u ) ( u ^{\frac{5}{7}} + 1 )}{u ^{\frac{4}{7}}} d u

Expand

\frac{( u ^{\frac{2}{7}} + 3 u ) ( u ^{\frac{5}{7}} + 1 )}{u ^{\frac{4}{7}}} : 4 u^{\frac{3}{7}} + 3 u^{\frac{8}{7}} + \frac{1}{u ^{\frac{2}{7}}}

= \frac{1}{14} \cdot \int 4 u^{\frac{3}{7}} + 3 u^{\frac{8}{7}} + \frac{1}{u ^{\frac{2}{7}}} d u

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}{14} (\int 4 u^{\frac{3}{7}} d u + \int 3 u^{\frac{8}{7}} d u + \int \frac{1}{u ^{\frac{2}{7}}} d u)

\int 4 u^{\frac{3}{7}} d u = \frac{14}{5} u^{\frac{10}{7}}

\int 3 u^{\frac{8}{7}} d u = \frac{7}{5} u^{\frac{15}{7}}

\int \frac{1}{u ^{\frac{2}{7}}} d u = \frac{7}{5} u^{\frac{5}{7}}

= \frac{1}{14} (\frac{14}{5} u^{\frac{10}{7}} + \frac{7}{5} u^{\frac{15}{7}} + \frac{7}{5} u^{\frac{5}{7}})

Substitute back

u = x^{14}

= \frac{1}{14} (\frac{14}{5} (x^{14})^{\frac{10}{7}} + \frac{7}{5} (x^{14})^{\frac{15}{7}} + \frac{7}{5} (x^{14})^{\frac{5}{7}})

Simplify

\frac{1}{14} (\frac{14}{5} (x^{14})^{\frac{10}{7}} + \frac{7}{5} (x^{14})^{\frac{15}{7}} + \frac{7}{5} (x^{14})^{\frac{5}{7}}) : \frac{1}{10} x^{10} (x^{20} + 2 x^{10} + 1)

= \frac{1}{10} x^{10} (x^{20} + 2 x^{10} + 1)

Add a constant to the solution

= \frac{1}{10} x^{10} (x^{20} + 2 x^{10} + 1) + C

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What is the integral of (3x^{14}+x^4)(x^{15}+x^5) ?
The integral of (3x^{14}+x^4)(x^{15}+x^5) is 1/10 x^{10}(x^{20}+2x^{10}+1)+C