Harder Substitution (Edexcel A-Level Mathematics): Quizzes

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Harder Substitution
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For ∫x1+x2 dx\int x \sqrt{1 + x^2} \, dx, what substitution should be used?

u=1+x2u = 1 + x^2

What is ∫x1+x2 dx\int x \sqrt{1 + x^2} \, dx equal to?

13(1+x2)32+C\frac{1}{3}(1 + x^2)^{\frac{3}{2}} + C

For ∫sin⁡(2x)1+cos⁡(2x) dx\int \frac{\sin(2x)}{1 + \cos(2x)} \, dx, what is the substitution?

u=1+cos⁡(2x)u = 1 + \cos(2x)

What is ∫xex2 dx\int x e^{x^2} \, dx equal to?

12ex2+C\frac{1}{2} e^{x^2} + C

For ∫ln⁡(x)x dx\int \frac{\ln(x)}{x} \, dx, what substitution is used?

u=ln⁡(x)u = \ln(x)

For ∫1x2x2−1 dx\int \frac{1}{x^2 \sqrt{x^2 - 1}} \, dx, which substitution is used?

x=sec⁡(θ)x = \sec(\theta)

Which identity is used when x2−1\sqrt{x^2 - 1} appears?

sec⁡2(θ)−1=tan⁡2(θ)\sec^2(\theta) - 1 = \tan^2(\theta)

What is ∫ln⁡(x)x dx\int \frac{\ln(x)}{x} \, dx equal to?

(ln⁡(x))22+C\frac{(\ln(x))^2}{2} + C

For ∫dx(1+x2)3/2\int \frac{dx}{(1 + x^2)^{3/2}}, which substitution is used?

x=tan⁡(θ)x = \tan(\theta)

Why is u=1+x2u = 1 + x^2 good for ∫x1+x2 dx\int x \sqrt{1 + x^2} \, dx?

Derivative 2x2x is almost present

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