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  4. Evaluate: ∫ (2x-3)5+(1/(7x-5)3)+(1/√5x-4)+(1/2-3x)+√3x+2 dx

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

Integrals

Solution:

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

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

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

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