First slide

Methods of integration

Question

$\int \frac{(x^{3} - 1)}{(x^{4} + 1) (x + 1)} d x =$

Moderate

A

$\frac{1}{4} \ln (1 + x^{4}) + \frac{1}{3} \ln (1 + x^{3}) + c$
B

$\frac{1}{4} \ln (1 + x^{4}) - \frac{1}{3} \ln (1 + x^{3}) + c$
C

$\frac{1}{4} \ln (1 + x^{4}) - \ln (1 + x) + c$
D

$\frac{1}{4} \ln (1 + x^{4}) + \ln (1 + x) + c$

Solution

$\begin{aligned} \int \frac{x^{3} - 1}{(x^{4} + 1) (x + 1)} dx = \int \frac{(x^{4} + x^{3}) - (x^{4} + 1)}{(x^{4} + 1) (x + 1)} dx \\ = \int \frac{x^{3}}{x^{4} + 1} dx - \int \frac{1}{x + 1} dx \\ = \frac{1}{4} \ln (x^{4} + 1) - \ln (x + 1) + c \end{aligned}$

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