Integral of e^x[f(x) + f'(x)]

The shortcut formula

Whenever an integrand has the exact shape $e^x$ times $[f(x) + f'(x)]$ , it integrates instantly, no substitution or parts needed:

\int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + C

This follows directly from the product rule run in reverse: $\frac{d}{dx}\left[e^xf(x)\right] = e^xf(x) + e^xf'(x) = e^x[f(x)+f'(x)]$ .

Worked example

Find $\displaystyle\int e^x\left(\dfrac{1}{x} - \dfrac{1}{x^2}\right)dx$ .

Solution: Let $f(x) = \dfrac{1}{x}$ . Then $f'(x) = -\dfrac{1}{x^2}$ , exactly matching the second term. So the integrand is $e^x[f(x)+f'(x)]$ , and:

\int e^x\left(\dfrac{1}{x} - \dfrac{1}{x^2}\right)dx = e^xf(x) + C = \dfrac{e^x}{x} + C

Integral of e^x[f(x) + f'(x)]

The shortcut formula

Whenever an integrand has the exact shape $e^x$ times $[f(x) + f'(x)]$ , it integrates instantly, no substitution or parts needed:

\int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + C

This follows directly from the product rule run in reverse: $\frac{d}{dx}\left[e^xf(x)\right] = e^xf(x) + e^xf'(x) = e^x[f(x)+f'(x)]$ .

Worked example

Find $\displaystyle\int e^x\left(\dfrac{1}{x} - \dfrac{1}{x^2}\right)dx$ .

Solution: Let $f(x) = \dfrac{1}{x}$ . Then $f'(x) = -\dfrac{1}{x^2}$ , exactly matching the second term. So the integrand is $e^x[f(x)+f'(x)]$ , and:

\int e^x\left(\dfrac{1}{x} - \dfrac{1}{x^2}\right)dx = e^xf(x) + C = \dfrac{e^x}{x} + C

Integral of ex[f(x) + f'(x)]

The shortcut formula

Worked example

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Integral of ex[f(x) + f'(x)]

The shortcut formula

Worked example

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Integral of e^x[f(x) + f'(x)]

Integral of e^x[f(x) + f'(x)]