Definite/indefinite integration (incl. 1/x, trig, e^x); substitution practice questions.

Determine ∫ 6x² dx.

• 2x³ + C
• 12x + C
• 6x³ + C
• 3x³ + C

Answer: 2x³ + C

Increase the power by 1 and divide by the new power: 6·x³/3 = 2x³ (+ C).

Determine ∫ (1/x) dx for x > 0.

• −1/x² + C
• x⁻² + C
• ln x + C
• ln|x²| + C

Answer: ln x + C

The integral of 1/x is ln|x| + C; for x > 0 this is ln x + C. Do not apply the power rule to x⁻¹.

Determine ∫ e^(3x) dx.

• 3e^(3x) + C
• e^(3x) + C
• (1/3)e^(3x) + C
• (1/3)e^(x) + C

Answer: (1/3)e^(3x) + C

∫ e^(ax) dx = (1/a)e^(ax) + C, so with a = 3 the result is (1/3)e^(3x) + C.

Determine ∫ sin(2x) dx.

• −(1/2)cos(2x) + C
• (1/2)cos(2x) + C
• −2cos(2x) + C
• 2cos(2x) + C

Answer: −(1/2)cos(2x) + C

∫ sin(ax) dx = −(1/a)cos(ax) + C. With a = 2 this gives −(1/2)cos(2x) + C.

Using the substitution u = x² + 1, determine ∫ 2x(x² + 1)⁴ dx.

• (1/5)(x² + 1)⁵ + C
• (1/4)(x² + 1)⁴ + C
• 5(x² + 1)⁵ + C
• (1/5)(x² + 1)⁴ + C

Answer: (1/5)(x² + 1)⁵ + C

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

Determine ∫ (4x³ − 2x + 5) dx.

• 12x² − 2 + C
• x⁴ − x² + 5x + C
• 4x⁴ − 2x² + 5x + C
• x⁴ − 2x² + 5 + C

Answer: x⁴ − x² + 5x + C

Integrate term by term: 4x⁴/4 − 2x²/2 + 5x = x⁴ − x² + 5x + C.