What is the integral of x\sqrt[6]{1+x^2} ?

The integral of x\sqrt[6]{1+x^2} is 3/7 (x^2+1)^{7/6}+C

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integral of x\sqrt[6]{1+x^2}

\int x 6 1 + x^{2} d x

\frac{3}{7} (x^{2} + 1)^{\frac{7}{6}} + C

Solution steps

\int x 6 1 + x^{2} d x

Apply u-substitution

: \int \frac{6 u}{2} d u

= \int \frac{6 u}{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 6 u d u

Apply the Power Rule

: \frac{6}{7} u^{\frac{7}{6}}

= \frac{1}{2} \cdot \frac{6}{7} u^{\frac{7}{6}}

Substitute back

u = 1 + x^{2}

= \frac{1}{2} \cdot \frac{6}{7} (1 + x^{2})^{\frac{7}{6}}

Simplify

\frac{1}{2} \cdot \frac{6}{7} (1 + x^{2})^{\frac{7}{6}} : \frac{3}{7} (x^{2} + 1)^{\frac{7}{6}}

= \frac{3}{7} (x^{2} + 1)^{\frac{7}{6}}

Add a constant to the solution

= \frac{3}{7} (x^{2} + 1)^{\frac{7}{6}} + C

\int \frac{cos ( \frac{π}{2} )}{x ^{2}} d x

\frac{d}{d x} (x^{2} (1 + y)^{3})

\frac{d}{d x} (cos^{l n (7 x)} (x))

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

\int (x)^{2} d x

What is the integral of x\sqrt[6]{1+x^2} ?
The integral of x\sqrt[6]{1+x^2} is 3/7 (x^2+1)^{7/6}+C