The rate of change of the volume of liquid, V V\,V in cm3\text{cm}^3cm3, in a hydraulic cylinder is modeled by the derivative dVdt=10t4−23t5+34\displaystyle \frac{dV}{dt} = 10t^4 - \frac{2}{3t^5} + \frac{3}{4}dtdV=10t4−3t52+43 for t>0t > 0t>0, where t t\,t is the time in seconds. Find the general expression for V V\,V by evaluating:
∫(10t4−23t5+34)dt \int \left( 10t^4 - \frac{2}{3t^5} + \frac{3}{4} \right) dt ∫(10t4−3t52+43)dtgiving each term in its simplest form.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.