Question

Using the Table of Integrals solve the following integrals:
(Make sure to state which equation you use)
a) ∫1/(25+x^2) dx
b) ∫x/((x+3)^2) dx

Answer 1

Answer on Question #52766 – Math – Integral Calculus

Using the Table of Integrals solve the following integrals:

(Make sure to state which equation you use)

a) $\int 1 / (25 + x^2) \, dx$

Solution

$$\int \frac {1}{25 + x ^ {2}} \, dx = \frac {1}{5} \arctan \frac {x}{5} + c,$$

where $c$ is an arbitrary real constant.

We used the following formula $\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan \frac{x}{a} + c,$

where $c$ is an arbitrary real constant.

b) $\int x / ((x + 3)^2) \, dx$

Solution

$$\begin{aligned} \int \frac {x}{(x + 3) ^ {2}} \, dx &= \int \frac {(x + 3) - 3}{(x + 3) ^ {2}} \, dx = \int \frac {(x + 3)}{(x + 3) ^ {2}} \, dx - \int \frac {3}{(x + 3) ^ {2}} \, dx = \int \frac {(x + 3) d (x + 3)}{(x + 3) ^ {2}} - 3 \int \frac {d (x + 3)}{(x + 3) ^ {2}} \\ &= |x + 3 - t| = \int \frac {t \, dt}{t ^ {2}} - 3 \int \frac {dt}{t ^ {2}} = \int \frac {dt}{t} + 3 \int \left(- \frac {1}{t ^ {2}}\right) dt = \ln |t| + \frac {3}{t} + C = \ln |x + 3| + \frac {3}{x + 3} + C, \end{aligned}$$

where $C$ is an arbitrary real constant.

We used the following formulae:

$$\int \frac {dt}{t} = \ln |t| + C,$$ $$\int t ^ {n} \, dt = \frac {t ^ {n + 1}}{n + 1} + C, \quad n \neq - 1,$$

where $C$ is an arbitrary real constant.

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