What is the integral of ln(8+2x) ?

The integral of ln(8+2x) is (ln(2(x+4))-1)(x+4)+C

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integral of ln(8+2x)

\int ln (8 + 2 x) d x

(ln (2 (x + 4)) - 1) (x + 4) + C

Solution steps

\int ln (8 + 2 x) d x

Apply u-substitution

: \int ln (u) \frac{1}{2} d u

= \int ln (u) \frac{1}{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 ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= \frac{1}{2} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{2} (u ln (u) - u)

Substitute back

u = 8 + 2 x

= \frac{1}{2} ((8 + 2 x) ln (8 + 2 x) - (8 + 2 x))

Simplify

\frac{1}{2} ((8 + 2 x) ln (8 + 2 x) - (8 + 2 x)) : (ln (2 (x + 4)) - 1) (x + 4)

= (ln (2 (x + 4)) - 1) (x + 4)

Add a constant to the solution

= (ln (2 (x + 4)) - 1) (x + 4) + C

\int (\frac{1}{4} x^{6} - 5 x^{3} + 9 x) d x

\frac{\partial}{\partial x} (- (xx - yy) e^{- xx - yy})

x \to \infty lim (\frac{e ^{x^{3}}}{7 - x ^{4}})

f^{^{'}} (x) = \frac{1}{x}

y = x^{3} - 2 x + 1

What is the integral of ln(8+2x) ?
The integral of ln(8+2x) is (ln(2(x+4))-1)(x+4)+C