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

The integral of ((t-3))/((t^2+7)) is 1/2 ln|t^2+7|-3/(sqrt(7))arctan(t/(sqrt(7)))+C

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integral of ((t-3))/((t^2+7))

integral

\frac{( t - 3 )}{( t ^{2} + 7 )}

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

Solution steps

\int \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 \frac{t}{t ^{2} + 7} d t - \int \frac{3}{t ^{2} + 7} d t

\int \frac{t}{t ^{2} + 7} d t = \frac{1}{2} ln t^{2} + 7

\int \frac{3}{t ^{2} + 7} d t = \frac{3}{7} arctan (\frac{t}{7})

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

Add a constant to the solution

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

e^{2 + x}

- 2 sec^{2} (x)

sin^{7} (x)

\frac{( x + 5 )}{( x ^{2} + 8 x + 17 )}

- 3 y^{2}

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