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Q.
- Rewrite Integral: Rewrite the integral by splitting the fraction into two parts.
- Simplify First Term: Simplify the first term by multiplying the numerator and denominator by to rationalize the denominator.
- Simplify Expression: Simplify the expression.\int\frac{x(\sqrt{x}+\(1\))}{x\(-1\)}\,dx = \int\frac{x^{\frac{\(3\)}{\(2\)}}+x}{x\(-1\)}\,dx
- Simplify Second Term: Now, let's simplify the second term by multiplying the numerator and denominator by \(\sqrt{x}+1.
- Combine Integrals: Combine the two integrals. \(\int\frac{x^{\frac{\(3\)}{\(2\)}}+x}{x\(-1\)}\,dx + \int\frac{\sqrt{x}+\(1\)}{x\(-1\)}\,dx
- Perform Integration: Perform long division or partial fraction decomposition for each term to integrate. This step is complex and requires careful algebraic manipulation.
