Integration of x upon 1 plus x sq (solve it by three methods) anyone please
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Answer:
Integrals Involving
√
a2−x2
Before developing a general strategy for integrals containing
√
a2−x2
, consider the integral ∫
√
9−x2
dx. This integral cannot be evaluated using any of the techniques we have discussed so far. However, if we make the substitution x=3sinθ, we have dx=3cosθdθ. After substituting into the integral, we have
∫
√
9−x2
dx=∫
√
9−(3sinθ)2
3cosθdθ.
After simplifying, we have
∫
√
9−x2
dx=∫9
√
1−sin2θ
cosθdθ.
Letting 1−sin2θ=cos2θ, we now have
∫
√
9−x2
dx=∫9
√
cos2θ
cosθdθ.
Assuming that cosθ≥0, we have
∫
√
9−x2
dx=∫9cos2θdθ.
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