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

The integral of (t^2+2)^3 is (t^7)/7+(6t^5)/5+4t^3+8t+C

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integral of (t^2+2)^3

\int (t^{2} + 2)^{3} d t

\frac{t ^{7}}{7} + \frac{6 t ^{5}}{5} + 4 t^{3} + 8 t + C

Solution steps

\int (t^{2} + 2)^{3} d t

Expand

(t^{2} + 2)^{3} : t^{6} + 6 t^{4} + 12 t^{2} + 8

= \int t^{6} + 6 t^{4} + 12 t^{2} + 8 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 t^{6} d t + \int 6 t^{4} d t + \int 12 t^{2} d t + \int 8 d t

\int t^{6} d t = \frac{t ^{7}}{7}

\int 6 t^{4} d t = \frac{6 t ^{5}}{5}

\int 12 t^{2} d t = 4 t^{3}

\int 8 d t = 8 t

= \frac{t ^{7}}{7} + \frac{6 t ^{5}}{5} + 4 t^{3} + 8 t

Add a constant to the solution

= \frac{t ^{7}}{7} + \frac{6 t ^{5}}{5} + 4 t^{3} + 8 t + C

\frac{d}{d t} (6 sin (t))

y = 4 x^{2}

x \to - 2 lim ((x^{2} - 1) (7 x + 3))

\frac{1}{s ^{2} ( s - 1 )}

\int (3 e^{x} + 5 sec^{2} (x)) d x

What is the integral of (t^2+2)^3 ?
The integral of (t^2+2)^3 is (t^7)/7+(6t^5)/5+4t^3+8t+C