5.4 Chain Rule in Reverse - Introduction to Calculus

Chain Rule in Reverse

The derivative of

\displaystyle f(u(x))

\displaystyle f'(u(x))u'(x)

, so

THEOREM

Chain Rule in Reverse

\displaystyle \begin{aligned}\int f'(u(x))u'(x) \ dx = f(u(x)) + C\end{aligned}

Notice that in the integration, the

\displaystyle u'(x)

piece disappears, being absorbed back into

\displaystyle f(u(x))

. The steps for finding the antiderivative of composite functions are as follows:

Identify the core layer $\displaystyle u(x)$ .

Identify the derivative of the core layer $\displaystyle u'(x)$ .

Identify the outer layer $\displaystyle f'$ , and integrate $\displaystyle f'$ leaving $\displaystyle u(x)$ inside.

Find the antiderivatives.

(a)

\displaystyle \int (6x^2 + 1)\sin(2x^3 + x) \ dx

(b)

\displaystyle \int \sec^2(4t) \ dt

(c)

\displaystyle \int 4x^3(3x^4 - 1)^{14} \ dx

(d)

\displaystyle \int {e^{1 \over x} \over 4x^2} \ dx

Integrate in one step.

\displaystyle \int e^{-2t} + \sin(3t) + \cos\left({1 \over 4}t\right) \ dt

QUESTION

Which expression equals

\displaystyle \int\sec^2(4t)\,dt

Loading lesson… / 正在打开教程…

Chain Rule in Reverse

The derivative of

\displaystyle f(u(x))

\displaystyle f'(u(x))u'(x)

, so

THEOREM

Chain Rule in Reverse

\displaystyle \begin{aligned}\int f'(u(x))u'(x) \ dx = f(u(x)) + C\end{aligned}

Notice that in the integration, the

\displaystyle u'(x)

piece disappears, being absorbed back into

\displaystyle f(u(x))

. The steps for finding the antiderivative of composite functions are as follows:

Identify the core layer $\displaystyle u(x)$ .

Identify the derivative of the core layer $\displaystyle u'(x)$ .

Identify the outer layer $\displaystyle f'$ , and integrate $\displaystyle f'$ leaving $\displaystyle u(x)$ inside.

Find the antiderivatives.

(a)

\displaystyle \int (6x^2 + 1)\sin(2x^3 + x) \ dx

(b)

\displaystyle \int \sec^2(4t) \ dt

(c)

\displaystyle \int 4x^3(3x^4 - 1)^{14} \ dx

(d)

\displaystyle \int {e^{1 \over x} \over 4x^2} \ dx

Integrate in one step.

\displaystyle \int e^{-2t} + \sin(3t) + \cos\left({1 \over 4}t\right) \ dt

QUESTION

Which expression equals

\displaystyle \int\sec^2(4t)\,dt

Chain Rule in Reverse

Chain Rule in Reverse

Chain Rule in Reverse

Reverse the chain rule

(a)

(b)

(c)

(d)

Integrate in one step

KNOWLEDGE QUIZ

Chain Rule in Reverse

Chain Rule in Reverse

Chain Rule in Reverse

Reverse the chain rule

(a)

(b)

(c)

(d)

Integrate in one step

KNOWLEDGE QUIZ