u-Substitution

College/University

Definition

An integration technique that can help make integration easier to carry out. It is used when an integral contains a function and it derivative. It involves using the chain rule in reverse.

Worked examples

\(\int 2x \cos(x^2) \, dx\)

Let \(u = x^2\), so \(du = 2x \, dx\). The integral becomes \(\int \cos(u) \, du = \sin(u) + C = \sin(x^2) + C\).

\(\int \frac{3x^2}{x^3 + 1} \, dx\)

Let \(u = x^3 + 1\), so \(du = 3x^2 \, dx\). Then \(\int \frac{1}{u} \, du = \ln|u| + C = \ln|x^3 + 1| + C\).

\(\int e^{5x} \, dx\)

Let \(u = 5x\), so \(du = 5 \, dx\) or \(dx = \frac{1}{5} du\). Then \(\frac{1}{5} \int e^u \, du = \frac{1}{5} e^{5x} + C\).

Common mistakes

\(\int 2x \cos(x^2) \, dx = 2x \sin(x^2) + C\) → \(\int 2x \cos(x^2) \, dx = \sin(x^2) + C\) You cannot integrate term-by-term when the function is composite; use u-substitution to handle the chain rule in reverse.
Let \(u = x^2\), then \(\int 2x \cos(x^2) \, dx = \int \cos(u) \, dx\) → Let \(u = x^2\), \(du = 2x \, dx\), then \(\int \cos(u) \, du\) After substituting u, you must also change dx to du — the entire differential must match your substitution.
\(\int x \cos(x^2) \, dx\) with \(u = x^2\), \(du = 2x \, dx\) gives \(\int \cos(u) \, du\) → \(du = 2x \, dx\), so \(x \, dx = \frac{1}{2} du\); the integral is \(\frac{1}{2} \int \cos(u) \, du\) When the derivative is off by a constant factor, solve for the correct differential and adjust the integral with that constant.

Where you'll use it next

u-Substitution is essential for more advanced integration techniques like integration by parts, trigonometric substitution, and solving differential equations in calculus II and beyond.

Found in 1 StudyPug lesson

U-Substitution: The Key to Solving Complex Integrals

Calculus 2

Unlock the power of U-Substitution in calculus. Learn to simplify and solve challenging integrals with our comprehensive guide, practice problems, and real-world applications.

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