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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u ^{\frac{3}{4}}}{2} d u

= \int \frac{u ^{\frac{3}{4}}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u^{\frac{3}{4}} d u

Apply the Power Rule

: \frac{4}{7} u^{\frac{7}{4}}

= \frac{1}{2} \cdot \frac{4}{7} u^{\frac{7}{4}}

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

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