Integration by substitution (simple) practice questions.

Which of the following is ∫ (1/x) dx?

• ln|x| + C
• −1/x² + C
• x + C
• 1/x + C

Answer: ln|x| + C

The integral of 1/x is the natural logarithm ln|x| + C.

Using the substitution u = x³ + 1, determine ∫ 6x²(x³ + 1)⁴ dx.

• (1/5)(x³ + 1)⁵ + C
• 2(x³ + 1)⁵ + C
• (2/5)(x³ + 1)⁵ + C
• (2/5)(x³ + 1)³ + C

Answer: (2/5)(x³ + 1)⁵ + C

With du = 3x² dx, the integral becomes 2∫u⁴ du = (2/5)u⁵ + C = (2/5)(x³ + 1)⁵ + C.

Determine ∫ x e^(x²) dx.

• e^(x²) + C
• (1/2)e^(x²) + C
• 2e^(x²) + C
• (1/2)x² e^(x²) + C

Answer: (1/2)e^(x²) + C

Let u = x², so du = 2x dx and the integral is (1/2)∫e^u du = (1/2)e^(x²) + C.

Using the substitution u = cos x, determine ∫ sin x cos⁴x dx.

• (1/5)cos⁵x + C
• (1/5)sin⁵x + C
• −(1/5)cos⁵x + C
• −(1/4)cos⁴x + C

Answer: −(1/5)cos⁵x + C

Since du = −sin x dx, the integral becomes −∫u⁴ du = −(1/5)u⁵ + C = −(1/5)cos⁵x + C; forgetting the minus sign is the classic error.

Using the substitution u = 2x + 1, determine ∫ (2x + 1)³ dx.

• (1/4)(2x + 1)⁴ + C
• (1/2)(2x + 1)⁴ + C
• 6(2x + 1)² + C
• (1/8)(2x + 1)⁴ + C

Answer: (1/8)(2x + 1)⁴ + C

du = 2 dx gives (1/2)∫u³ du = (1/8)u⁴ + C; omitting the factor 1/2 from dx = du/2 leads to the common wrong answer (1/4)(2x + 1)⁴.

Determine ∫ x/(x² + 4) dx.

• ln(x² + 4) + C
• (1/2)ln(x² + 4) + C
• (1/2)arctan(x/2) + C
• 2 ln(x² + 4) + C

Answer: (1/2)ln(x² + 4) + C

With u = x² + 4, du = 2x dx, so the integral is (1/2)∫(1/u) du = (1/2)ln(x² + 4) + C; the arctan form applies to 1/(x² + 4), not x/(x² + 4).