 New Zealand
Level 8 - NCEA Level 3

Indefinite and definite integrals (e^(f(x) non-linear)

Lesson

Any integral that can be expressed in the form $\int\ f'(t).g\left(f(t)\right)\ \mathrm{d}t$ f(t).g(f(t)) dt has an antiderivative $G\left(f(t)\right)$G(f(t)) where the derivative of $G$G is $g$g.

This comes from the procedure for differentiating a function of a function.

Example 1

In particular, if $G$G is the exponential function, $G(x)=e^x$G(x)=ex, we could have a function $H(x)=G\left(f(x)\right)$H(x)=G(f(x)) given by $e^{x^2}$ex2. Then $H'(x)$H(x), the derivative, is $2x.e^{x^2}$2x.ex2.

Going the other way, if we were given an antiderivative $\int\ 2x.e^{x^2}\ \mathrm{d}x$ 2x.ex2 dx, we should recognise the form $f'(x).e^{f(x)}$f(x).ef(x) and immediately write down the simplified antiderivative $e^{x^2}+C$ex2+C.

Example 2

Simplify $\int\ \cos x.e^{\sin x}\ \mathrm{d}x$ cosx.esinx dx.

Because $\cos x$cosx is the derivative of $\sin x$sinx, we see that $\int\ \cos x.e^{\sin x}\ \mathrm{d}x=e^{\sin x}+C$ cosx.esinx dx=esinx+C.

Example 3

Evaluate $I=\int_0^2\ e^{e^t+t}\ \mathrm{d}t$I=20 eet+t dt.

Using an index law, we have

 $I$I $=$= $\int_0^2\ e^{e^t+t}\ \mathrm{d}t$∫20​ eet+t dt $=$= $\int_0^2\ e^t.e^{e^t}\ \mathrm{d}t$∫20​ et.eet dt $=$= $\left[e^{e^t}\right]_0^2$[eet]20​ $=$= $e^{e^2}-e$ee2−e $\approx$≈ $1615.5$1615.5

Worked examples

Question 1

Consider the function $y=e^{x^3}$y=ex3.

1. Find the derivative $\frac{dy}{dx}$dydx.

2. Hence find the value of $\int3x^2e^{x^3}dx$3x2ex3dx.

You may use $C$C to represent the constant of integration.

Question 2

Determine the value of $\int\left(4x^3+3\right)e^{x^4+3x}dx$(4x3+3)ex4+3xdx.

1. You may use $C$C to represent the constant of integration.

Question 3

Find the value of $I$I, where $I=\int_{-1}^1\left(4x^3-3x^2\right)e^{x^4-x^3}dx$I=11(4x33x2)ex4x3dx.

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