What is the integral from cos(x) to pi of (t+1)^2 ?

The integral from cos(x) to pi of (t+1)^2 is (pi^3-cos^3(x))/3+pi^2-cos^2(x)+pi-cos(x)

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integral from cos(x) to pi of (t+1)^2

\int_{c o s (x)}^{π} (t + 1)^{2} d t

\frac{π ^{3} - cos ^{3} ( x )}{3} + π^{2} - cos^{2} (x) + π - cos (x)

Solution steps

\int_{c o s (x)}^{π} (t + 1)^{2} d t

Expand

(t + 1)^{2} : t^{2} + 2 t + 1

= \int_{c o s (x)}^{π} t^{2} + 2 t + 1 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_{c o s (x)}^{π} t^{2} d t + \int_{c o s (x)}^{π} 2 t d t + \int_{c o s (x)}^{π} 1 d t

\int_{c o s (x)}^{π} t^{2} d t = \frac{π ^{3} - cos ^{3} ( x )}{3}

\int_{c o s (x)}^{π} 2 t d t = π^{2} - cos^{2} (x)

\int_{c o s (x)}^{π} 1 d t = π - cos (x)

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

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What is the integral from cos(x) to pi of (t+1)^2 ?
The integral from cos(x) to pi of (t+1)^2 is (pi^3-cos^3(x))/3+pi^2-cos^2(x)+pi-cos(x)