What is the integral of f(ax+b) ?

The integral of f(ax+b) is f(bx+(ax^2)/2)+C

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integral of f(ax+b)

\int f (a x + b) d x

f (b x + \frac{a x ^{2}}{2}) + C

Solution steps

\int f (a x + b) d x

Take the constant out:

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

= f \cdot \int b + a 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

= f (\int b d x + \int a x d x)

\int b d x = b x

\int a x d x = \frac{a x ^{2}}{2}

= f (b x + \frac{a x ^{2}}{2})

Add a constant to the solution

= f (b x + \frac{a x ^{2}}{2}) + C

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