MHT CET · Maths · Indefinite Integration
\(\int(3-x) \sqrt{4-x} d x=\) (Where \(C\) is a constant of integration.)
- A \(\frac{2}{3}(4-x)^{3 / 2}+\frac{2}{5}(4-x)^{5 / 2}+C\)
- B \(-\frac{2}{5}(4-x)^{5 / 2}+\frac{2}{3}(4-x)^{3 / 2}+C\)
- C \(\frac{2}{3}(4-x)^{3 / 2}-\frac{2}{5}(4-x)^{5 / 2}+C\)
- D \(\frac{2}{5}(4-x)^{5 / 2}-\frac{2}{5}(4-x)^{3 / 2}+C\)
Answer & Solution
Correct Answer
(C) \(\frac{2}{3}(4-x)^{3 / 2}-\frac{2}{5}(4-x)^{5 / 2}+C\)
Step-by-step Solution
Detailed explanation
\(\int(3-x) \sqrt{4-x} d x\)
\(=\int\{(4-x)-1\} \sqrt{4-x} d x=\int\{(4-x)^{\frac{3}{2}}-\) \((4-x)^{\frac{1}{2}}\} d x\)
\(=\frac{2}{5}(4-x)^{5 / 2}+\frac{2}{3}(4-x)^{3 / 2}+C\)
\(=\frac{2}{3}(4-x)^{3 / 2}-\frac{2}{5}(4-x)^{5 / 2}+C\)
\(=\int\{(4-x)-1\} \sqrt{4-x} d x=\int\{(4-x)^{\frac{3}{2}}-\) \((4-x)^{\frac{1}{2}}\} d x\)
\(=\frac{2}{5}(4-x)^{5 / 2}+\frac{2}{3}(4-x)^{3 / 2}+C\)
\(=\frac{2}{3}(4-x)^{3 / 2}-\frac{2}{5}(4-x)^{5 / 2}+C\)
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