DSAverse

Initializing Sorting Algorithms...

Sorting

Trees

Graphs

Preparing interactive visualizations...

Back to Integral Calculus

Integration by Parts

Draw the curve (u, v). The rectangle under its end point splits into two regions — ∫ u dv on one side, ∫ v du on the other — and that is the whole formula.

∫ x cos x dx · step 1 of 20

— the curve (u, v) = (x, sin x)■ ∫ v du■ ∫ u dv

Speed

∫ x cos x dx: a product, and no substitution untangles it. Integration by parts picks u = x and dv = cos x dx, so du = dx and v = sin x. Now draw the curve traced by (u, v) = (x, sin x) as x runs from 0 to π/2.

u = x,\quad v = \sin x,\quad \text{curve } (u, v) \text{ for } x \text{ from } 0 \text{ to } 1.5708

\int_0^{1.5708} x\cos x\,dx =\int u\,dv = \big[uv\big] - \int v\,du = 1.5708 - 1 = 0.5708

Integration by Parts

Draw the curve (u, v). The rectangle under its end point splits into two regions — ∫ u dv on one side, ∫ v du on the other — and that is the whole formula.

∫ x cos x dx · step 1 of 20

— the curve (u, v) = (x, sin x)■ ∫ v du■ ∫ u dv

Speed

u = x,\quad v = \sin x,\quad \text{curve } (u, v) \text{ for } x \text{ from } 0 \text{ to } 1.5708

\int_0^{1.5708} x\cos x\,dx =\int u\,dv = \big[uv\big] - \int v\,du = 1.5708 - 1 = 0.5708

DSAverse

Integration by Parts

See also

Integration by Parts

See also