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Find $$\int \frac{x}{4+x^2} \ dx.$$ - HSC - SSCE Mathematics Advanced - Question 17 - 2020 - Paper 1

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Find-$$\int-\frac{x}{4+x^2}-\-dx.$$-HSC-SSCE Mathematics Advanced-Question 17-2020-Paper 1.png

Find $$\int \frac{x}{4+x^2} \ dx.$$

Worked Solution & Example Answer:Find $$\int \frac{x}{4+x^2} \ dx.$$ - HSC - SSCE Mathematics Advanced - Question 17 - 2020 - Paper 1

Step 1

Step 1: Rewrite the Integral

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Answer

The integral can be rewritten as: ∫x4+x2 dx=∫12⋅2x4+x2 dx\int \frac{x}{4+x^2} \ dx = \int \frac{1}{2} \cdot \frac{2x}{4+x^2} \ dx.

Step 2

Step 2: Use Substitution

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Answer

Letting ( u = 4 + x^2 ), then ( du = 2x \ dx ). Thus, we can rewrite the integral as: 12∫1u du.\frac{1}{2} \int \frac{1}{u} \ du. This simplifies to: 12ln⁡∣u∣+C=12ln⁡∣4+x2∣+C.\frac{1}{2} \ln |u| + C = \frac{1}{2} \ln |4+x^2| + C.

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