Question

$\int \frac{{(x+a)}^3}{x^3}+\frac{1-x^4}{1-x}-\frac{7}{x\sqrt{x^2-1}}+\frac{x+2}{{(x+1)}^2}+\frac{{(1+x)}^3}{\sqrt{x}}dx$ is equal to

• A. $x+3a\log x-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7{\sec }^{-1}x+\log (x+1)-\frac{1}{x+1}+2\sqrt{x}+2x^{\frac{3}{2}}+\frac{6}{5}{x}^{\frac{5}{2}}+\frac{2}{7}{x}^{\frac{7}{2}}+C$
• B. $x+2a\log x-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7{\sec }^{-1}x+\log (x+1)-\frac{1}{x+1}+2\sqrt{x}+2x^{\frac{3}{2}}+\frac{6}{5}{x}^{\frac{5}{2}}+\frac{2}{7}{x}^{\frac{7}{2}}+C$
• C. $x+3a\log x-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7{\sec }^{-1}x+\log (x+1)-\frac{1}{x+1}+2\sqrt{x}+2x^{\frac{3}{2}}+\frac{6}{5}{x}^{\frac{5}{2}}+\frac{2}{7}{x}^{\frac{5}{2}}+C$
• D. none of these

Answer 1

The correct option is A $x+3a\log x-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7{\sec }^{-1}x+\log (x+1)-\frac{1}{x+1}+2\sqrt{x}+2x^{\frac{3}{2}}+\frac{6}{5}{x}^{\frac{5}{2}}+\frac{2}{7}{x}^{\frac{7}{2}}+C$

$\int \{\frac{{(x+a)}^3}{x^3}+\frac{1-x^4}{1-x}-\frac{7}{x\sqrt{x^2-1}}+\frac{x+2}{{(x+1)}^2}+\frac{{(1+x)}^3}{\sqrt{x}}\}dx$
.
$=\int \frac{x^3+3a{x}^2+3a^{2}x+a^3}{x^3}dx+\int \frac{(1-x)(1+x+x^2+x^3)}{(1-x)}dx-\int \frac{7}{x\sqrt{x^2-1}}dx+\int \frac{(x+1)}{{(x+1)}^2}dx+\int \frac{1}{{(x+1)}^2}dx+\int \frac{1+3x+3x^2+x^3}{x^{\frac{1}{2}}}dx$
.
Using integration of each function separately, we get
$=(x+3a\log x-\frac{3a^2}{x}-\frac{a^3}{2x^2})+(x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4})-\left(7{\sec }^{-1}x\right)+\left(\log (x+1)\right)(-\frac{1}{x+1})+(2\sqrt{x}+2x^{\frac{3}{2}}+\frac{6}{5}{x}^{\frac{5}{2}}+\frac{2}{7}{x}^{\frac{7}{2}})+C$