What is the integral from 0 to x of (t-3)/(t^2+7) ?

The integral from 0 to x of (t-3)/(t^2+7) is 1/2 (ln|x^2+7|-ln(7))-3/(sqrt(7))arctan(x/(sqrt(7)))

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integral from 0 to x of (t-3)/(t^2+7)

\int_{0}^{x} \frac{t - 3}{t ^{2} + 7} d t

\frac{1}{2} (ln x^{2} + 7 - ln (7)) - \frac{3}{7} arctan (\frac{x}{7})

Solution steps

\int_{0}^{x} \frac{t - 3}{t ^{2} + 7} d t

Expand

\frac{t - 3}{t ^{2} + 7} : \frac{t}{t ^{2} + 7} - \frac{3}{t ^{2} + 7}

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}^{x} \frac{t}{t ^{2} + 7} d t - \int_{0}^{x} \frac{3}{t ^{2} + 7} d t

\int_{0}^{x} \frac{t}{t ^{2} + 7} d t = \frac{1}{2} (ln x^{2} + 7 - ln (7))

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

= \frac{1}{2} (ln x^{2} + 7 - ln (7)) - \frac{3}{7} arctan (\frac{x}{7})

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What is the integral from 0 to x of (t-3)/(t^2+7) ?
The integral from 0 to x of (t-3)/(t^2+7) is 1/2 (ln|x^2+7|-ln(7))-3/(sqrt(7))arctan(x/(sqrt(7)))