Integration by parts

Japanese school year: Math III

What you learn

This section covers integration by parts, a technique derived from the product rule of differentiation. It is widely used to integrate products of polynomials, exponential, and trigonometric functions. A solid understanding of the product rule and basic indefinite integration will help you master this method.

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Key points

This integration-by-parts formula is used to integrate the product of two functions. By differentiating $f(x)$ into $f'(x)$ and integrating $g'(x)$ into $g(x)$ , it converts a difficult integral into a simpler one.

\int f(x)g'(x) \, dx = f(x)g(x) - \int f'(x)g(x) \, dx

This formula applies integration by parts to definite integrals. You evaluate the product $f(x)g(x)$ between the upper limit $b$ and lower limit $a$ , then subtract the remaining definite integral.

\int_{a}^{b} f(x)g'(x) \, dx = \left[ f(x)g(x) \right]_{a}^{b} - \int_{a}^{b} f'(x)g(x) \, dx

This guideline helps choose roles for functions in integration by parts. Set $f(x)$ as the function that simplifies upon differentiation, and $g'(x)$ as the function that is straightforward to integrate.

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Practice 1Coming soon