10 questions🇬🇧English

∫ Integration by Parts & Substitution – A-level

Maths · A-level — practise choosing u and dv/dx, substitutions, changing limits and recognising standard integral forms.

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Questions

Which expression correctly completes the integration by parts formula ∫u (dv/dx) dx = … ?

uv − ∫v (du/dx) dx

uv − ∫u (dv/dx) dx

u(dv/dx) − ∫v (du/dx) dx

uv + ∫v (du/dx) dx

20s

To find ∫x eˣ dx by parts, which choice of u and dv/dx works best?

u = x, dv/dx = eˣ

u = x eˣ, dv/dx = 1

u = eˣ, dv/dx = x

u = 1, dv/dx = x eˣ

20s

Find ∫ 2x ÷ (x² + 1) dx.

−1 ÷ (x² + 1) + c

2 ln(x² + 1) + c

½ ln(x² + 1) + c

ln(x² + 1) + c

20s

Using integration by parts, find ∫x sin x dx.

−x cos x − sin x + c

x cos x − sin x + c

x cos x + sin x + c

−x cos x + sin x + c

30s

Using u = ln x and dv/dx = 1, integration by parts gives ∫ln x dx equal to:

x ln x + c

x ln x − x + c

x ln x + x + c

1/x + c

30s

Using the reverse chain rule, find ∫x(x² + 3)⁴ dx.

(1/5)(x² + 3)⁵ + c

2(x² + 3)⁵ + c

(1/2)(x² + 3)⁵ + c

(1/10)(x² + 3)⁵ + c

30s

Which is the most efficient method to find ∫x e^(x²) dx?

Parts with u = e^(x²), dv/dx = x

Substitution u = x²

Partial fractions

Parts with u = x, dv/dx = e^(x²)

30s

Using u = x² + 1, the integral ∫₀¹ 2x(x² + 1)³ dx becomes:

∫₁² u³ du

∫₁² 2x u³ du

∫₀² u³ du

∫₀¹ u³ du

30s

Using integration by parts twice, find ∫x² eˣ dx.

eˣ(x² − 2x − 2) + c

eˣ(x² − 2x + 2) + c

eˣ(x² + 2x + 2) + c

eˣ(x² − 2x) + c

60s

Evaluate ∫₀^(π/2) x cos x dx.

1 − π/2

π/2

π/2 + 1

π/2 − 1

60s