H: Integration: Integrate xⁿ (excluding n = −1), and related sums, differences and constant multiples. Integrate e^(kx), 1/x, sin kx, cos kx and related sums, differences and constant multiples.

The idea

This section builds the core toolkit for indefinite integration. For powers, you reverse differentiation: raise the index by one and divide by the new index, so ∫xⁿ dx = xⁿ⁺¹/(n+1) + c, valid for any constant n except n = −1. The exception is handled by ∫(1/x) dx = ln|x| + c. You also integrate exponential and trigonometric functions: ∫e^(kx) dx = (1/k)e^(kx) + c, ∫sin kx dx = −(1/k)cos kx + c, and ∫cos kx dx = (1/k)sin kx + c. These rules extend to sums, differences and constant multiples by integrating term by term and taking constants outside the integral. Always include the constant of integration for indefinite integrals.

For example: ∫(3x² + 2e^(2x) − 5/x) dx = x³ + e^(2x) − 5 ln|x| + c.

What earns the marks

• Correctly applies the power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1, including fractional and negative indices.
• Recognises that ∫(1/x) dx = ln|x| + c and does not incorrectly apply the power rule to n = −1.
• Integrates exponential functions using ∫e^(kx) dx = (1/k)e^(kx) + c, including the reciprocal factor 1/k.
• Integrates trigonometric functions with correct signs and reciprocal factors: ∫sin kx dx = −(1/k)cos kx + c and ∫cos kx dx = (1/k)sin kx + c.
• Handles sums, differences and constant multiples by integrating term by term and factoring out constants.
• Includes the constant of integration, c, for indefinite integrals.

Common mistakes and how to put them right

• Forgetting to divide by the new index when integrating powers, e.g. writing ∫x³ dx = x⁴ + c instead of x⁴/4 + c.

  Put it right: Divide by (n+1).

• Applying the power rule to 1/x, giving x⁰/0, which is undefined.

  Put it right: Use ln|x| for the integral of 1/x.

• Omitting the constant of integration, c, in indefinite integrals.

  Put it right: Add c unless evaluating a definite integral.

• Incorrect sign for ∫sin kx dx, writing +(1/k)cos kx instead of −(1/k)cos kx.

  Put it right: The derivative of cos is −sin, so the integral of sin is −cos.

• Forgetting the reciprocal factor 1/k when integrating e^(kx), sin kx or cos kx.

  Put it right: Divide by the coefficient of x.

Exam tips

• Write the constant of integration, c, clearly in every indefinite integral answer; it is often worth a mark.
• For terms like 1/x, rewrite as x⁻¹ only if you remember the exception; otherwise go straight to ln|x|.
• Check your answer by differentiating it back to the original integrand; this catches sign and factor errors quickly.
• When integrating sums, split the integral into separate terms and deal with each one carefully, especially signs.