What is the integral of (8x^3+4x)(x^4+x^2)^{5/2} ?

The integral of (8x^3+4x)(x^4+x^2)^{5/2} is 4/7 x^7(x^2+1)^{7/2}+C

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integral of (8x^3+4x)(x^4+x^2)^{5/2}

\int (8 x^{3} + 4 x) (x^{4} + x^{2})^{\frac{5}{2}} d x

\frac{4}{7} x^{7} (x^{2} + 1)^{\frac{7}{2}} + C

Solution steps

\int (8 x^{3} + 4 x) (x^{4} + x^{2})^{\frac{5}{2}} d x

Apply u-substitution

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

Take the constant out:

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

= 2 \cdot \int u^{\frac{5}{2}} d u

Apply the Power Rule

= 2 \cdot \frac{2}{7} u^{\frac{7}{2}}

Substitute back

u = x^{4} + x^{2}

= 2 \cdot \frac{2}{7} (x^{4} + x^{2})^{\frac{7}{2}}

Simplify

2 \cdot \frac{2}{7} (x^{4} + x^{2})^{\frac{7}{2}} : \frac{4}{7} x^{7} (x^{2} + 1)^{\frac{7}{2}}

= \frac{4}{7} x^{7} (x^{2} + 1)^{\frac{7}{2}}

Add a constant to the solution

= \frac{4}{7} x^{7} (x^{2} + 1)^{\frac{7}{2}} + C

What is the integral of (8x^3+4x)(x^4+x^2)^{5/2} ?
The integral of (8x^3+4x)(x^4+x^2)^{5/2} is 4/7 x^7(x^2+1)^{7/2}+C