What is the integral of (2x^3+x)(x^4+x^2)^{108} ?

The integral of (2x^3+x)(x^4+x^2)^{108} is 1/218 (x^4+x^2)^{109}+C

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integral of (2x^3+x)(x^4+x^2)^{108}

\int (2 x^{3} + x) (x^{4} + x^{2})^{108} d x

\frac{1}{218} (x^{4} + x^{2})^{109} + C

Solution steps

\int (2 x^{3} + x) (x^{4} + x^{2})^{108} d x

Apply u-substitution

: \int \frac{u ^{108}}{2} d u

= \int \frac{u ^{108}}{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^{108} d u

Apply the Power Rule

: \frac{u ^{109}}{109}

= \frac{1}{2} \cdot \frac{u ^{109}}{109}

Substitute back

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

= \frac{1}{2} \cdot \frac{( x ^{4} + x ^{2} ) ^{109}}{109}

Simplify

\frac{1}{2} \cdot \frac{( x ^{4} + x ^{2} ) ^{109}}{109} : \frac{1}{218} (x^{4} + x^{2})^{109}

= \frac{1}{218} (x^{4} + x^{2})^{109}

Add a constant to the solution

= \frac{1}{218} (x^{4} + x^{2})^{109} + C

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\frac{d}{d x} (\frac{24}{x ^{2} + 4 x})

sin (2 t) sin (t)

What is the integral of (2x^3+x)(x^4+x^2)^{108} ?
The integral of (2x^3+x)(x^4+x^2)^{108} is 1/218 (x^4+x^2)^{109}+C