AP EAMCET · Maths · Definite Integration
If \(f(x)=\frac{x^3+5}{\sqrt{12+x}}\) and \(\int_{-5}^5 f(x) d x=\int_0^5(f(x)+g(x)) d x\), then \(g(x)=\)
- A \(\frac{5-x^3}{\sqrt{12-x}}\)
- B \(-\left(\frac{5+\mathrm{x}^3}{\sqrt{12+x}}\right)\)
- C \(\frac{-\mathrm{x}^3+5}{\sqrt{12+\mathrm{x}}}\)
- D \(\frac{5+x^3}{\sqrt{12-x}}\)
Answer & Solution
Correct Answer
(A) \(\frac{5-x^3}{\sqrt{12-x}}\)
Step-by-step Solution
Detailed explanation
Let \(I=\int_{-5}^5 \frac{x^3+5}{\sqrt{12+x}} d x...(1)\) \[ \Rightarrow I=\int_{-5}^5 \frac{(5-5-x)^3+5}{\sqrt{12+(5-5-x)}} d x...(By Property)...(2) \] Adding equation (1) \& (2) \[ 2 I=\int_{-5}^5\left(\frac{x^3+5}{\sqrt{12+x}}+\frac{-x^3+5}{\sqrt{12-x}}\right) d x \] Since…
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