What is the integral from 0 to 7 of (x^4+5)/(1+x^2) ?

The integral from 0 to 7 of (x^4+5)/(1+x^2) is 6arctan(7)+322/3

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integral from 0 to 7 of (x^4+5)/(1+x^2)

\int_{0}^{7} \frac{x ^{4} + 5}{1 + x ^{2}} d x

6 arctan (7) + \frac{322}{3}

Decimal

115.90672 \dots

Solution steps

\int_{0}^{7} \frac{x ^{4} + 5}{1 + x ^{2}} d x

Expand

\frac{x ^{4} + 5}{1 + x ^{2}} : \frac{x ^{4}}{1 + x ^{2}} + \frac{5}{1 + x ^{2}}

Apply the Sum Rule:

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

= \int_{0}^{7} \frac{x ^{4}}{1 + x ^{2}} d x + \int_{0}^{7} \frac{5}{1 + x ^{2}} d x

\int_{0}^{7} \frac{x ^{4}}{1 + x ^{2}} d x = arctan (7) + \frac{322}{3}

\int_{0}^{7} \frac{5}{1 + x ^{2}} d x = 5 arctan (7)

= arctan (7) + \frac{322}{3} + 5 arctan (7)

Simplify

= 6 arctan (7) + \frac{322}{3}

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What is the integral from 0 to 7 of (x^4+5)/(1+x^2) ?
The integral from 0 to 7 of (x^4+5)/(1+x^2) is 6arctan(7)+322/3