What is the integral of (14x+1)sqrt(7x^2+x) ?

The integral of (14x+1)sqrt(7x^2+x) is 2/3 (7x^2+x)^{3/2}+C

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integral of (14x+1)sqrt(7x^2+x)

\int (14 x + 1) 7 x^{2} + x d x

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

Solution steps

\int (14 x + 1) 7 x^{2} + x d x

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 7 x^{2} + x

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

Add a constant to the solution

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

x^{2} \frac{d y}{d x} + x (x + 2) y = e^{x}

\int \frac{7}{( x - 3 ) ( x + 4 )} d x

e^{- \frac{t}{4}}

t \to 1 lim (e^{4 t})

\frac{d y}{d x} - 4 y = x^{6} e^{x}

What is the integral of (14x+1)sqrt(7x^2+x) ?
The integral of (14x+1)sqrt(7x^2+x) is 2/3 (7x^2+x)^{3/2}+C