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)

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

$\Rightarrow I=\int {\left(2x-3\right)}^{5}dx+\int {\left(7x-5\right)}^{-3}dx+\int {\left(5x-4\right)}^{-\frac{1}{2}}dx+\int \frac{1}{2-3x}dx+\int \sqrt{3x+2}dx$

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

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