What is the integral of (x+2)^7 ?

The integral of (x+2)^7 is x(x+2)^7-(7x^8)/8-12x^7-70x^6-224x^5-420x^4-448x^3-224x^2+C

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integral of (x+2)^7

\int (x + 2)^{7} d x

x (x + 2)^{7} - \frac{7 x ^{8}}{8} - 12 x^{7} - 70 x^{6} - 224 x^{5} - 420 x^{4} - 448 x^{3} - 224 x^{2} + C

Solution steps

\int (x + 2)^{7} d x

Apply Integration By Parts

: (x + 2)^{7} x - \int 7 (x + 2)^{6} x d x

= (x + 2)^{7} x - \int 7 (x + 2)^{6} x d x

\int 7 (x + 2)^{6} x d x = \frac{7 x ^{8}}{8} + 12 x^{7} + 70 x^{6} + 224 x^{5} + 420 x^{4} + 448 x^{3} + 224 x^{2}

= (x + 2)^{7} x - (\frac{7 x ^{8}}{8} + 12 x^{7} + 70 x^{6} + 224 x^{5} + 420 x^{4} + 448 x^{3} + 224 x^{2})

Simplify

(x + 2)^{7} x - (\frac{7 x ^{8}}{8} + 12 x^{7} + 70 x^{6} + 224 x^{5} + 420 x^{4} + 448 x^{3} + 224 x^{2}) : x (x + 2)^{7} - \frac{7 x ^{8}}{8} - 12 x^{7} - 70 x^{6} - 224 x^{5} - 420 x^{4} - 448 x^{3} - 224 x^{2}

= x (x + 2)^{7} - \frac{7 x ^{8}}{8} - 12 x^{7} - 70 x^{6} - 224 x^{5} - 420 x^{4} - 448 x^{3} - 224 x^{2}

Add a constant to the solution

= x (x + 2)^{7} - \frac{7 x ^{8}}{8} - 12 x^{7} - 70 x^{6} - 224 x^{5} - 420 x^{4} - 448 x^{3} - 224 x^{2} + C

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What is the integral of (x+2)^7 ?
The integral of (x+2)^7 is x(x+2)^7-(7x^8)/8-12x^7-70x^6-224x^5-420x^4-448x^3-224x^2+C