Integration by U-Substitution - Calculus Step by Step

Find ∫2x · cos(x²) dx by substituting u = x², du = 2x dx, giving sin(x²) + C.

u-substitution U-substitution reverses the chain rule: let u = g(x), du = g'(x) dx, then ∫f(g(x))g'(x) dx = ∫f(u) du. Back-substitute u = g(x) to get the final answer.

Example

Choose u, convert to du, integrate in u, and back-substitute.

highlighted = computed this step

Step 1 — Set up

Set up the composite integral.

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

Step 2 — Choose u

Let u equal the inside function.

u= \hlmath{x^{2}}

Step 3 — Compute du

Differentiate u to get du.

du= \hlmath{2x} \,dx

Step 4 — Integrate in u

Integrate cosine u to get sine u plus C.

\int \cos u \,du= \hlmath{\sin u} +C

Step 5 — Back-substitute

Back-substitute x squared for u.

\hlmath{\sin(x^{2})} +C