2.5 Integration

Use substitution when an inner derivative is present, and integration by parts ∫u dv=uv−∫v du when a product becomes simpler after differentiating one factor.

For substitution, change the differential and limits consistently. For parts, choose u to become simpler and dv to be easy to integrate; differentiate the final answer to verify.

∫2x cos(x²)dx=sin(x²)+C by u=x². For ∫x e^x dx, take u=x and dv=e^x dx.

Substitution is not complete until every x and dx is replaced, and integration by parts does not remove the need for C in an indefinite integral.

Rewrite powers or products using identities such as sin²x=(1−cos2x)/2, then integrate term by term. If an inner angle is present, substitute it with its differential.

Use the identity that reduces the power or creates a derivative factor; keep absolute values in logarithmic antiderivatives such as ∫tan x dx=−ln|cos x|+C.

∫sin²x dx=x/2−sin2x/4+C, while ∫sec²(3x)dx=tan(3x)/3+C.

∫sin²x is not −cos³x/3; the power is on the function, not on its differential pattern.

With equal spacing h and ordinates y₀,…,yₙ, the trapezium estimate is h/2[y₀+yₙ+2(y₁+…+yₙ₋₁)]. It approximates the definite integral by straight chords.

Use n subintervals and h=(b−a)/n. For a curve that is concave up, the chord estimate lies above or below according to the curvature, so compare with an exact value when available.

Using h=0.5 with ordinates at x=0,0.5,1 gives area ≈0.25[y₀+2y₁+y₂].

The number of ordinates is n+1, not n, and the endpoint weights are 1 rather than 2.