Integration and the area under a curve

Learning objectives

Understand integration as the reverse of differentiation, and include an arbitrary constant in every indefinite integral

Integrate sums of terms in powers of x, including x⁻¹ which gives ln|x|, and (ax + b)ⁿ for any rational n other than −1, together with 1/(ax + b), e^(ax + b), sin(ax + b), cos(ax + b) and sec²(ax + b), remembering to divide by a

Find the equation of a curve from its gradient function and a point that lies on it, by integrating and using the point to determine the constant

Evaluate a definite integral by substituting the limits into the antiderivative and subtracting, and use the result to find the area between a curve and the x-axis

Find the area between a curve and the y-axis by integrating with respect to y, and the area between two curves by integrating the difference after finding the points of intersection

Handle a region that lies partly below the x-axis by splitting the integral at the roots and taking the magnitude of each part, explaining why a single integral would give the wrong area