Quick questions on Integration: areas, standard integrals, substitution and by parts - Edexcel A-Level Maths

A rational function whose denominator factorises can be split into simpler fractions, each of which integrates to a logarithm. This is the standard route for integrands such as $\dfrac{1}{(x - a)(x - b)}$.

The area between a curve and the $x$-axis is $\int_a^b y\,dx$. Where the curve dips below the axis the integral is negative, so split the region at the roots and add the magnitudes. The area between two curves $y = f(x)$ and $y = g(x)$ with $f \ge g$ is $\int_a^b (f(x) - g(x))\,dx$.

A separable equation $\dfrac{dy}{dx} = f(x)g(y)$ is solved by writing $\int \dfrac{1}{g(y)}\,dy = \int f(x)\,dx$ and integrating each side, then using a boundary condition to find the constant. Such equations model situations where the rate of change depends on the current value, such as cooling, population growth and the discharge of a capacitor.

Pick $u$ as the factor that simplifies on differentiating, often a polynomial.

Find $\int (6x^2 - 4x + 1)\,dx$. [2 marks]

Evaluate $\int_1^2 \dfrac{1}{x}\,dx$. [3 marks]