# Integration and Differentiation Practice Questions

Try this selection of differentiation and integration questions to practise choosing an appropriate technique.

## 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 $x$):

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:

- $\frac{1}{4}\left( x+9 \right)^4 + c$
- $-\cos x \ln(\cos x)+\cos x + c$
- $\frac{2}{3}(1+x^2)^{3/2}+c$
- $-e^{\cos x}+c$
- $-x^2 \cos x + 2x \sin x + 2 \cos x + c$
- $\frac{1}{2}e^x (\cos x + \sin x) + c$