$\int \left(x^{3/2}+2e^{x}-\dfrac{1}{x}\right)dx$ equals
A$\dfrac{5}{2}x^{5/2}+e^{x}-\log|x|+C$
B$\dfrac{2}{5}x^{5/2}+2e^{x}+\log|x|+C$
C$\dfrac{3}{2}x^{1/2}+2e^{x}-\log|x|+C$
D$\dfrac{2}{5}x^{5/2}+2e^{x}-\log|x|+C$
Answer & Solution
Correct answer: D. $\dfrac{2}{5}x^{5/2}+2e^{x}-\log|x|+C$
1. $\int x^{3/2}\,dx=\dfrac{x^{5/2}}{5/2}=\dfrac{2}{5}x^{5/2}$.
2. $\int 2e^{x}\,dx=2e^{x}$.
3. $\int \dfrac{1}{x}\,dx=\log|x|$, subtracted gives $-\log|x|$.
4. Sum: $\dfrac{2}{5}x^{5/2}+2e^{x}-\log|x|+C$. Option B has the wrong sign on the log term.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.294_
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