What is the integral of xsqrt(x)(1+x^2sqrt(x))^4 ?

The integral of xsqrt(x)(1+x^2sqrt(x))^4 is 2/25 (1+x^2sqrt(x))^5+C

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integral of xsqrt(x)(1+x^2sqrt(x))^4

\int x x (1 + x^{2} x)^{4} d x

\frac{2}{25} (1 + x^{2} x)^{5} + C

Solution steps

\int x x (1 + x^{2} x)^{4} d x

Apply u-substitution

: \int \frac{2 u ^{4}}{5} d u

= \int \frac{2 u ^{4}}{5} d u

Take the constant out:

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

= \frac{2}{5} \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

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

Substitute back

u = 1 + x^{2} x

= \frac{2}{5} \cdot \frac{( 1 + x ^{2} x ) ^{5}}{5}

Simplify

\frac{2}{5} \cdot \frac{( 1 + x ^{2} x ) ^{5}}{5} : \frac{2}{25} (1 + x^{2} x)^{5}

= \frac{2}{25} (1 + x^{2} x)^{5}

Add a constant to the solution

= \frac{2}{25} (1 + x^{2} x)^{5} + C

y^{3} - (6 x + 6) + 3 x y^{2} y^{^{'}} = 0

\frac{d y}{d x} = \frac{x ^{2}}{y ^{2}}

\int 7 x^{5} d x

x \to 0 + lim (tan^{2 x} (x))

e^{5 t} sin (2 t)

What is the integral of xsqrt(x)(1+x^2sqrt(x))^4 ?
The integral of xsqrt(x)(1+x^2sqrt(x))^4 is 2/25 (1+x^2sqrt(x))^5+C