6 0 | ( sin 3 x + sin 2 x + sin x ) | ? d x is equal to_____. [2026]

Question

$6 \int_{0}^{π}$ $| (\sin 3 x + \sin 2 x + \sin x) |$ $d x$ is equal to_____. [2026]

Smart Achievers · Accepted Answer

(17)

$6 \int_{0}^{π}$ $| 2 \sin 2 x \cos x + \sin 2 x |$ $d x$

$= 6 \int_{0}^{π}$ $| 4 \sin x \cos^{2} x + 2 \sin x \cos x |$ $d x$

$I = 12 \int_{0}^{π}$ $| \sin x (2 \cos^{2} x + \cos x) |$ $d x$

Put $\cos x = t, - \sin x d x = d t$

$I = - 12 \int_{1}^{- 1}$ $| 2 t^{2} + t |$ $d t$

$I = 12 (\int_{- 1}^{- \frac{1}{2}} (2 t^{2} + t) d t + \int_{- \frac{1}{2}}^{0} - (2 t^{2} + t) d t + \int_{0}^{1} (2 t^{2} + t) d t)$

$I = 17$

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