Question

help please. I will be very thankful to u

Answer 1

$\bold{Answer :}$

Let us take, x = a tanθ

∴ dx = a sec²θ dθ

Then,

x² + a²

= a² tan²θ + a²

= a² (tan²θ + 1)

= a² sec²θ [ ∵ sec²θ - tan²θ = 1 ]

Now,

1/(x² + a²)

= 1/(a² sec²θ)

∴ ∫ 1/(x² + a²) dx

= ∫ (a sec²θ dθ)/(a² sec²θ)

= ∫ (dθ)/a

= 1/a ∫ dθ

= (1/a) θ + c, where c is integral constant

= (1/a) tan⁻¹(x/a) + c [ ∵ x = a tanθ ]

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