What is the integral of-7v^2+7 v/x+7+3/x ?

The integral of-7v^2+7 v/x+7+3/x is 7x+3ln|x|+7vln|x|-7v^2x+C

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integral of-7v^2+7 v/x+7+3/x

\int - 7 v^{2} + 7 \frac{v}{x} + 7 + \frac{3}{x} d x

7 x + 3 ln ∣ x ∣ + 7 v ln ∣ x ∣ - 7 v^{2} x + C

Solution steps

\int - 7 v^{2} + 7 \cdot \frac{v}{x} + 7 + \frac{3}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 7 d x + \int \frac{3}{x} d x + \int 7 \cdot \frac{v}{x} d x - \int 7 v^{2} d x

\int 7 d x = 7 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

\int 7 \cdot \frac{v}{x} d x = 7 v ln ∣ x ∣

\int 7 v^{2} d x = 7 v^{2} x

= 7 x + 3 ln ∣ x ∣ + 7 v ln ∣ x ∣ - 7 v^{2} x

Add a constant to the solution

= 7 x + 3 ln ∣ x ∣ + 7 v ln ∣ x ∣ - 7 v^{2} x + C

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What is the integral of-7v^2+7 v/x+7+3/x ?
The integral of-7v^2+7 v/x+7+3/x is 7x+3ln|x|+7vln|x|-7v^2x+C