Integration by Substitution: Step-by-Step Guide
u-substitution is the integral analog of the chain rule: if an integrand contains a function and a multiple of its derivative, we set u equal to the inner function so that du captures the derivative factor. This converts the integral into a simpler one in the variable u that we can solve with basic rules.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Choose u. Pick u to be an inner function whose derivative also appears (up to a constant) in the integrand. Common choices: the inside of a composition, the radicand, or the exponent.\[u = g(x)\]
- Compute du. Differentiate u with respect to x and solve for dx (or for the relevant factor). Adjust by constants if needed.\[du = g'(x)\,dx \;\Rightarrow\; dx = \frac{du}{g'(x)}\]
- Rewrite the integral in terms of u. Substitute u and du into the integrand. Every x must disappear; if it doesn't, your choice of u was wrong.\[\int f(g(x))\,g'(x)\,dx = \int f(u)\,du\]
- Integrate with respect to u. Apply standard rules (power, exponential, trig) to integrate f(u) du.\[\int f(u)\,du = F(u) + C\]
- Back-substitute. Replace u with g(x) so the answer is in the original variable. Verify by differentiating.\[F(u) + C = F(g(x)) + C\]
Worked example
Evaluate the integral of 2x*cos(x^2) dx.
Step 1: Notice the inner function is x^2 and its derivative 2x is already present. Let u = x^2.
Step 2: Then du = 2x dx, so 2x dx = du.
Step 3: Rewrite the integral as integral of cos(u) du.
Step 4: Integrate: integral of cos(u) du = sin(u) + C.
Step 5: Back-substitute u = x^2 to get sin(x^2) + C.
Verification: Differentiate sin(x^2) + C using the chain rule: cos(x^2) * 2x = 2x cos(x^2). Matches the original integrand.
Tips and common mistakes
- Look for a composition f(g(x)) where g'(x) (or a constant multiple) is already a factor in the integrand.
- Constants can be moved outside the integral; if du = 2x dx but you only have x dx, write x dx = (1/2) du.
- Every x must be replaced; if some x's remain after substitution, try a different u or solve for x in terms of u.
- For definite integrals, change the limits to u-values to avoid back-substitution.
- Common u choices: inside of trig functions, the exponent of e, denominators, and radicands.
- Always verify by differentiating your answer; substitution errors are easy to catch this way.
- If a single substitution doesn't simplify the integral, consider whether integration by parts or a trig identity is needed first.