Key Idea / Trick

Integration by parts:

\int u \, dv = uv - \int v \, du

Take

u = x^2

dv = \sin(x)\,dx

. Differentiate

u

and integrate

dv

repeatedly until the polynomial factor vanishes (requires 2 rounds). The tabular method organizes this cleanly.

Integral of $x^2 \sin(x)$ — Answer

Answer

$\int x^2 \sin(x)\, dx = 2x\sin(x) - (x^2 - 2)\cos(x) + C$

Solution via Tabular Method

Set up the table: differentiate the polynomial column until it hits 0, integrate the trig column, alternate signs $+, -, +, \ldots$

| Sign | Differentiate | Integrate | |------|--------------|-----------| | $+$ | $x^2$ | $\sin(x)$ | | $-$ | $2x$ | $-\cos(x)$ | | $+$ | $2$ | $-\sin(x)$ | | $-$ | $0$ | $\cos(x)$ |

Multiply diagonally (each row's sign × left entry × right entry of next row) and sum:

$= (+)(x^2)(-\cos x) + (-)(2x)(-\sin x) + (+)(2)(\cos x)$ $= -x^2\cos(x) + 2x\sin(x) + 2\cos(x) + C$

Factoring: $= 2x\sin(x) - (x^2 - 2)\cos(x) + C$

Verification

Differentiate the answer: $\frac{d}{dx}\left[2x\sin x - (x^2-2)\cos x\right]$ $= 2\sin x + 2x\cos x - 2x\cos x + (x^2-2)\sin x$ $= 2\sin x + (x^2-2)\sin x = x^2 \sin x \checkmark$

Integral of $x^2 \sin(x)$

Problem

Field

Why It's Interesting

Key Idea / Trick

Difficulty

Tags

Integral of $x^2 \sin(x)$ — Answer

Answer

Solution via Tabular Method

Verification

Integral of x2sin⁡(x)x^2 \sin(x)x2sin(x)

Problem

Field

Why It's Interesting

Key Idea / Trick

Difficulty

Tags

Integral of x2sin⁡(x)x^2 \sin(x)x2sin(x) — Answer

Answer

Solution via Tabular Method

Verification

Integral of $x^2 \sin(x)$

Integral of $x^2 \sin(x)$ — Answer