New Zealand
Level 8 - NCEA Level 3

# Finding primitive functions, indefinite integrals, definite integrals (mixed functions)

Lesson

The comments made on this topic in another chapter are also pertinent here.

In this chapter, we consider functions constructed from simpler functions, including the trigonometric functions, by addition and by multiplication by a constant.

We have seen that the derivative of a sum of functions is the sum of the derivatives of the functions. It follows that an antiderivative of a sum of functions is the sum of the individual antiderivatives.

It is easily verified with the product rule that the derivative of a constant multiple of a function is the constant multiplied by the derivative of the function. This also works in reverse so that the antiderivative $\int af(x)\ \mathrm{d}x$af(x) dxis the same as $a\int f(x)\ \mathrm{d}x$af(x) dx.

##### Example 1

Find the primitive $\int2\left(t+\sin t\right)\ \mathrm{d}t$2(t+sint) dt.

To begin, we can move the constant through the integral sign and write $2\int\left(t+\sin t\right)\ \mathrm{d}t$2(t+sint) dt.

This is the same as $2\left[\int t\ \mathrm{d}t+\int\sin t\ \mathrm{d}t\right]$2[t dt+sint dt]. So, the primitive is $t^2-2\cos t+C$t22cost+C.

##### Example 2

Evaluate the definite integral $\int_0^{\frac{\pi}{6}}\tan^2t\ \mathrm{d}t$π60tan2t dt.

To find an antiderivative for $\tan^2t$tan2t, we manipulate the integrand into a form made up of recognised derivatives. Now, $\tan^2t=\frac{\sin^2t}{\cos^2t}=\frac{1-\cos^2t}{\cos^2t}=\sec^2t-1$tan2t=sin2tcos2t=1cos2tcos2t=sec2t1.

So, we require the antiderivative $\int\left(\sec^2t-1\right)\ \mathrm{d}t$(sec2t1) dt. This is, $\tan t-t+C$tantt+C. Thus, we calculate $\left[\tan t-t\right]_0^{\frac{\pi}{6}}$[tantt]π60. Then,

 $\int_0^{\frac{\pi}{6}}\tan^2t\ \mathrm{d}t$∫π6​0​tan2t dt $=$= $\left[\tan t-t\right]_0^{\frac{\pi}{6}}$[tant−t]π6​0​ $=$= $\tan\frac{\pi}{6}-\frac{\pi}{6}$tanπ6​−π6​ $=$= $\frac{1}{\sqrt{3}}-\frac{\pi}{6}$1√3​−π6​

#### More Examples

##### Question 1

Find the integral of $\sin\left(\frac{x}{3}\right)$sin(x3).

You may use $C$C as the constant of integration.

##### Question 2

Determine $\int\left(\cos x-\sin3x\right)dx$(cosxsin3x)dx.

You may use $C$C as the constant of integration.

##### Question 3

Evaluate $\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(4\cos x+\cos4x\right)dx$π6π6(4cosx+cos4x)dx.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91579

Apply integration methods in solving problems