What is the integral of 3cos(pi-t)+(pi-t)^2 ?

The integral of 3cos(pi-t)+(pi-t)^2 is 3(tcos(pi-t)+tcos(t)-sin(t))+pi^2t-pit^2+(t^3)/3+C

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integral of 3cos(pi-t)+(pi-t)^2

\int 3 cos (π - t) + (π - t)^{2} d t

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

Solution steps

\int 3 cos (π - t) + (π - t)^{2} d t

π - t = π (1 - \frac{t}{π})

= \int 3 cos (π (1 - \frac{t}{π})) + (π - t)^{2} d t

π - t = π (1 - \frac{t}{π})

= \int 3 cos (π (1 - \frac{t}{π})) + (π (1 - \frac{t}{π}))^{2} d t

Apply the Sum Rule:

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

= \int 3 cos (π (1 - \frac{t}{π})) d t + \int (π (1 - \frac{t}{π}))^{2} d t

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

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

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

Add a constant to the solution

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

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What is the integral of 3cos(pi-t)+(pi-t)^2 ?
The integral of 3cos(pi-t)+(pi-t)^2 is 3(tcos(pi-t)+tcos(t)-sin(t))+pi^2t-pit^2+(t^3)/3+C