Question

Evaluate: \int\left\\{\left(2x-3\right)^{5}+\frac{1}{\left(7x-5\right)^{3}}+\frac{1}{\sqrt{5x-4}}+\frac{1}{2-3x}+\sqrt{3x+2}\right\\}dx

• A. $I= \frac{1}{12}\left(2x-3\right)^{6}-\frac{1}{14}\left(7x-5\right)^{-2}+\frac{2}{5}\sqrt{5x-4}-\frac{1}{3}log\left|2-3x\right|+\frac{2}{9}\left(3x+2\right)^{\frac{3}{2}}+C$
• B. $I= \frac{1}{12}\left(2x-3\right)^{6}-\frac{1}{14}\left(7x-5\right)^{-2}+\frac{2}{5}\sqrt{5x-4}$
• C. $I= \frac{1}{12}\left(2x-3\right)^{4}+\frac{1}{14}\left(7x-5\right)^{-2}$
• D. None of these

Answer 1

Answer: (A)

$I= \frac{1}{12}\left(2x-3\right)^{6}-\frac{1}{14}\left(7x-5\right)^{-2}+\frac{2}{5}\sqrt{5x-4}-\frac{1}{3}log\left|2-3x\right|+\frac{2}{9}\left(3x+2\right)^{\frac{3}{2}}+C$

Answer 2

\int\left\\{\left(2x-3\right)^{5}+\frac{1}{\left(7x-5\right)^{3}}+\frac{1}{\sqrt{5x-4}}+\frac{1}{2-3x}+\sqrt{3x+2}\right\\}dx $\Rightarrow I= \int\left(2x-3\right)^{5}dx +\int\left(7x-5\right)^{-3}dx +\int \left(5x-4\right)^{-\frac{1}{2}}dx +\int\frac{1}{2-3x}dx +\int \sqrt{3x+2}dx$ $\Rightarrow I = \frac{\left(2x-3\right)^{6}}{2\times6} + \frac{\left(7x - 5\right)^{-2}}{7\times\left(-2\right)} + \frac{\left(5x-4\right)^{\frac{1}{2}}}{5\times\frac{1}{2}} + \left(\frac{1}{-3}\right)log \left|2-3x\right| + \frac{\left(3x+2\right)^{\frac{3}{2}}}{3\times\frac{3}{2}}+C$ $\Rightarrow I = \frac{1}{12} \left(2x-3\right)^{6} - \frac{1}{14} \left(7x-5\right)^{-2}+\frac{2}{5}\sqrt{5x - 4}-\frac{1}{3}log \left|2-3x\right| + \frac{2}{9}\left(3x+2\right)^{\frac{3}{2}} + C$