What is the integral of (1-cos(2t))(cos(t))t ?

The integral of (1-cos(2t))(cos(t))t is 2(1/3 tsin^3(t)-1/3 (-cos(t)+(cos^3(t))/3))+C

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integral of (1-cos(2t))(cos(t))t

\int (1 - cos (2 t)) (cos (t)) t d t

2 (\frac{1}{3} t sin^{3} (t) - \frac{1}{3} (- cos (t) + \frac{cos ^{3} ( t )}{3})) + C

Solution steps

\int (1 - cos (2 t)) cos (t) t d t

Rewrite using trig identities

= \int 2 t sin^{2} (t) cos (t) d t

Take the constant out:

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

= 2 \cdot \int t sin^{2} (t) cos (t) d t

Apply Integration By Parts

: \frac{1}{3} t sin^{3} (t) - \int \frac{1}{3} sin^{3} (t) d t

= 2 (\frac{1}{3} t sin^{3} (t) - \int \frac{1}{3} sin^{3} (t) d t)

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

= 2 (\frac{1}{3} t sin^{3} (t) - \frac{1}{3} (- cos (t) + \frac{cos ^{3} ( t )}{3}))

Add a constant to the solution

= 2 (\frac{1}{3} t sin^{3} (t) - \frac{1}{3} (- cos (t) + \frac{cos ^{3} ( t )}{3})) + C

n = 0 \sum \infty n^{2} e^{- 7 n}

\int cos^{3} (X) sin^{2} (X) D X d X

\int 2 cos (x) - 2 sin (x) d x

\int - 2 x cos (8 x) d x

3 x - 5

What is the integral of (1-cos(2t))(cos(t))t ?
The integral of (1-cos(2t))(cos(t))t is 2(1/3 tsin^3(t)-1/3 (-cos(t)+(cos^3(t))/3))+C