Integration by substitution

Japanese school year: Math III

What you learn

Learn integration by substitution, which rewrites an integral by replacing part of the expression with a new variable and using derivative relations. This technique is widely used for solving differential equations and calculating complex areas and volumes. Prior knowledge of the chain rule and basic indefinite integrals is required.

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Key points

This substitution formula helps integrate complicated functions by changing variables. By setting part of the expression as $x = g(t)$ , the factor $g'(t) \, dt$ transforms into $dx$ , simplifying the integral.

\int f(g(t))g'(t) \, dt = \int f(x) \, dx \quad (x = g(t))

This formula applies substitution to definite integrals. When changing the variable from $t$ to $x = g(t)$ , the integration limits must also change from $a, b$ to $g(a), g(b)$ .

\int_{a}^{b} f(g(t))g'(t) \, dt = \int_{g(a)}^{g(b)} f(x) \, dx \quad (x = g(t))

This formula quickly integrates a fraction whose numerator $f'(x)$ is the derivative of its denominator $f(x)$ . The result directly evaluates to the natural logarithm $\log |f(x)|$ plus an integration constant $C$ .

\int \frac{f'(x)}{f(x)} \, dx = \log |f(x)| + C

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Practice 1Coming soon