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Evaluate the following. ∫1a2-b2x2 dx - Mathematics and Statistics

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Sum

Evaluate the following.

`int 1/("a"^2 - "b"^2 "x"^2)` dx

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Solution

Let I = `int 1/("a"^2 - "b"^2 "x"^2)` dx

`= 1/"b"^2 int 1/("a"^2/"b"^2 - "x"^2)`dx

`= 1/"b"^2 int 1/(("a"/"b")^2 - "x"^2)` dx

`= 1/"b"^2 xx 1/(2("a"/"b")) log |("a"/"b" + "x")/("a"/"b" - "x")|` + c

∴ I = `1/"2ab" log |("a" + "bx")/("a" - "bx")|` + c

Alternate Method:

Let I = `int "dx"/("a"^2 - "b"^2"x"^2) = int"dx"/("a"^2 - ("bx")^2)`

`= 1/(2 xx "a") xx 1/"b" log |("a" + "bx")/("a" - "bx")|` + c

∴ I = `1/"2ab" log |("a" + "bx")/("a" - "bx")|` + c

Notes

[Note: Answer in the textbook is incorrect.]

Concept: Methods of Integration: Integration by Substitution
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