What is the integral of 27x^7+3x^5-45x^3+sqrt(2)x ?

The integral of 27x^7+3x^5-45x^3+sqrt(2)x is (27x^8)/8+(x^6)/2-(45x^4)/4+(x^2)/(sqrt(2))+C

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integral of 27x^7+3x^5-45x^3+sqrt(2)x

\int 27 x^{7} + 3 x^{5} - 45 x^{3} + 2 x d x

\frac{27 x ^{8}}{8} + \frac{x ^{6}}{2} - \frac{45 x ^{4}}{4} + \frac{x ^{2}}{2} + C

Solution steps

\int 27 x^{7} + 3 x^{5} - 45 x^{3} + 2 x d x

Apply the Sum Rule:

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

= \int 27 x^{7} d x + \int 3 x^{5} d x - \int 45 x^{3} d x + \int 2 x d x

\int 27 x^{7} d x = \frac{27 x ^{8}}{8}

\int 3 x^{5} d x = \frac{x ^{6}}{2}

\int 45 x^{3} d x = \frac{45 x ^{4}}{4}

\int 2 x d x = \frac{x ^{2}}{2}

= \frac{27 x ^{8}}{8} + \frac{x ^{6}}{2} - \frac{45 x ^{4}}{4} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{27 x ^{8}}{8} + \frac{x ^{6}}{2} - \frac{45 x ^{4}}{4} + \frac{x ^{2}}{2} + C

What is the integral of 27x^7+3x^5-45x^3+sqrt(2)x ?
The integral of 27x^7+3x^5-45x^3+sqrt(2)x is (27x^8)/8+(x^6)/2-(45x^4)/4+(x^2)/(sqrt(2))+C