What is the integral from 0 to pi of 10tsin(3t) ?

The integral from 0 to pi of 10tsin(3t) is (10pi)/3

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integral from 0 to pi of 10tsin(3t)

\int_{0}^{π} 10 t sin (3 t) d t

\frac{10 π}{3}

Decimal

10.47197 \dots

Solution steps

\int_{0}^{π} 10 t sin (3 t) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int_{0}^{π} t sin (3 t) d t

Apply Integration By Parts

= 10 [\frac{1}{3} (- t cos (3 t) - 3 \cdot \int - \frac{1}{3} cos (3 t) d t)]_{0}^{π}

\int - \frac{1}{3} cos (3 t) d t = - \frac{1}{9} sin (3 t)

= 10 [\frac{1}{3} (- t cos (3 t) - 3 (- \frac{1}{9} sin (3 t)))]_{0}^{π}

Simplify

10 [\frac{1}{3} (- t cos (3 t) - 3 (- \frac{1}{9} sin (3 t)))]_{0}^{π} : 10 [\frac{1}{3} (- t cos (3 t) + \frac{1}{3} sin (3 t))]_{0}^{π}

= 10 [\frac{1}{3} (- t cos (3 t) + \frac{1}{3} sin (3 t))]_{0}^{π}

Compute the boundaries:

\frac{π}{3}

= 10 \cdot \frac{π}{3}

Simplify

= \frac{10 π}{3}

x \to \infty lim (\frac{x \cdot ( x + 1 ) ^{x + 1}}{( x + 1 ) ^{x + 2}})

\int (sin (x))^{2} (cos (x))^{2} d x

s l o p e in t erce pt (2005, 54), (2010, 58)

\int u^{2} + 1 d u

d er i v a t i v e ln (1 - 9 x)

What is the integral from 0 to pi of 10tsin(3t) ?
The integral from 0 to pi of 10tsin(3t) is (10pi)/3