What is the integral of (x^2-5)^20.2x ?

The integral of (x^2-5)^20.2x is 0.2(0.16666…x^6-2.5x^4+12.5x^2)+C

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integral of (x^2-5)^20.2x

\int (x^{2} - 5)^{2} 0.2 x d x

0.2 (0.16666 \dots x^{6} - 2.5 x^{4} + 12.5 x^{2}) + C

Solution steps

\int (x^{2} - 5)^{2} \cdot 0.2 x d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 0.2 \cdot \int (x^{2} - 5)^{2} x d x

Expand

(x^{2} - 5)^{2} x : x^{5} - 10 x^{3} + 25 x

= 0.2 \cdot \int x^{5} - 10 x^{3} + 25 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

= 0.2 (\int x^{5} d x - \int 10 x^{3} d x + \int 25 x d x)

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

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

\int 25 x d x = \frac{25 x ^{2}}{2}

= 0.2 (\frac{x ^{6}}{6} - \frac{5 x ^{4}}{2} + \frac{25 x ^{2}}{2})

Simplify

= 0.2 (0.16666 \dots x^{6} - 2.5 x^{4} + 12.5 x^{2})

Add a constant to the solution

= 0.2 (0.16666 \dots x^{6} - 2.5 x^{4} + 12.5 x^{2}) + C

\int_{6}^{7} \frac{x}{x - 6} d x

\int (3 x + 2) d x

\int - 1 e^{x} d x

\int \frac{x ^{3}}{3} + 2 x - 5 d x

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

What is the integral of (x^2-5)^20.2x ?
The integral of (x^2-5)^20.2x is 0.2(0.16666…x^6-2.5x^4+12.5x^2)+C