Integration and Differentiation Practice Questions
Try this selection of differentiation and integration questions to practise choosing an appropriate technique.
Age
16 to 18
Challenge level
Problem
There are a wide variety of techniques that can be used to solve differentiation and integration problems, such as the chain rule, the product rule, the quotient rule, integration by substitution, integration by parts. For each question below, think for a while about which technique is likely to be fruitful before diving in!
Differentiate the following expressions (with respect to $x$):
Integrate the following expressions (with respect to $x$):
You can generate more questions to practise different calculus techniques using the Mathmo app.
Differentiate the following expressions (with respect to $x$):
- $(8x+7) e^x$
- $\ln(6x^2+2x+5)$
- $\sin(-5x^2-8x+2)$
- $(-7x^2-5x) \cos x$
- $e^{x^2-8x+7}$
- $(4x^2-3x-7) \ln x$
Integrate the following expressions (with respect to $x$):
- $\left( x+9 \right)^3$
- $\sin x \ln(\cos x)$
- $2x\sqrt{1+x^2}$
- $e^{\cos x} \sin x$
- $x^2 \sin x$
- $e^x \cos x$
You can generate more questions to practise different calculus techniques using the Mathmo app.
Student Solutions
Differentiation:
Integration:
- $8 e^x + (8x+7) e^x$
- $\displaystyle\frac{12x+2}{6x^2+2x+5}$
- $(-10x-8)\cos(5x^2+8x-2)$
- $(-14x-5)\cos x + (7x^2+5x) \sin x$
- $\left( 2x-8 \right) e^{x^2-8x+7}$
- $(8x-3) \ln x + \displaystyle\frac{4x^2-3x-7}{x}$
Integration:
- $\frac{1}{4}\left( x+9 \right)^4 + c$
- $-\cos x \ln(\cos x)+\cos x + c$
- $\frac{2}{3}(1+x^2)^{3/2}+c$
- $-e^{\cos x}+c$
- $-x^2 \cos x + 2x \sin x + 2 \cos x + c$
- $\frac{1}{2}e^x (\cos x + \sin x) + c$