What is the integral from 1 to 1/x of cos^3(u) ?

The integral from 1 to 1/x of cos^3(u) is sin(1/x)-sin(1)-(sin^3(1/x)-sin^3(1))/3

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integral from 1 to 1/x of cos^3(u)

\int_{1}^{\frac{1}{x}} cos^{3} (u) d u

sin (\frac{1}{x}) - sin (1) - \frac{sin ^{3} ( \frac{1}{x} ) - sin ^{3} ( 1 )}{3}

Solution steps

\int_{1}^{\frac{1}{x}} cos^{3} (u) d u

Rewrite using trig identities

= \int_{1}^{\frac{1}{x}} (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

= \int_{s i n (1)}^{s i n (\frac{1}{x})} 1 - v^{2} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{s i n (1)}^{s i n (\frac{1}{x})} 1 d v - \int_{s i n (1)}^{s i n (\frac{1}{x})} v^{2} d v

\int_{s i n (1)}^{s i n (\frac{1}{x})} 1 d v = sin (\frac{1}{x}) - sin (1)

\int_{s i n (1)}^{s i n (\frac{1}{x})} v^{2} d v = \frac{sin ^{3} ( \frac{1}{x} ) - sin ^{3} ( 1 )}{3}

= sin (\frac{1}{x}) - sin (1) - \frac{sin ^{3} ( \frac{1}{x} ) - sin ^{3} ( 1 )}{3}

What is the integral from 1 to 1/x of cos^3(u) ?
The integral from 1 to 1/x of cos^3(u) is sin(1/x)-sin(1)-(sin^3(1/x)-sin^3(1))/3