Recognising Reverse Chain Rule | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Use a derivative you already know

A question saying hence integrate asks you to use the preceding result. If differentiating $F(x)$ gave the required integrand, its integral is $F(x)+c$ . A constant multiplier can be adjusted.

Worked example

Differentiate (x² + 4)⁴, then find ∫6x(x² + 4)³ dx.

\begin{gathered}\frac{d}{dx}(x^2+4)^4\\=4(x^2+4)^3\times2x\\=8x(x^2+4)^3\end{gathered}

The requested integrand is $6/8=3/4$ of that derivative. Multiply the original function by the same constant.

\begin{gathered}\int6x(x^2+4)^3\,dx\\=\frac34(x^2+4)^4+c\end{gathered}

Match the variable factor, not just the bracket

Without a supplied derivative, identify the inner function $g(x)$ and calculate $g'(x)$ . A power of the inner function can be integrated this way when the remaining factor is a constant multiple of $g'(x)$ . Differentiate the proposed answer to confirm the match.

For $\int(x^2+1)^2\,dx$ , the needed factor $x$ is absent. Do not divide the answer by $2x$ : that is a changing factor and creates extra derivative terms. Expanding the square gives the usable form $x^4+2x^2+1$ .

Your turnChoose the method

Find both integrals: $\int3x(2x^2+1)^4\,dx$ and $\int(x^2+1)^2\,dx$ .

Show worked answer

The first has the inner derivative factor: differentiating $(2x^2+1)^5$ produces $20x(2x^2+1)^4$ . The second needs expansion.

Answer

\begin{gathered}\frac3{20}(2x^2+1)^5+c\\\frac{x^5}{5}+\frac{2x^3}{3}+x+c\end{gathered}

These are separate indefinite integrals; their arbitrary constants need not have the same value.

Your turnPractice

Differentiate $(x^2+3)^{-1}$ . Hence find $\int4x/(x^2+3)^2\,dx$ .

Show worked answer

\begin{gathered}\frac{d}{dx}(x^2+3)^{-1}\\=-2x(x^2+3)^{-2}\end{gathered}

The requested integrand is −2 times this derivative.

Answer

-\frac2{x^2+3}+c

Use a derivative you already know

A question saying hence integrate asks you to use the preceding result. If differentiating $F(x)$ gave the required integrand, its integral is $F(x)+c$ . A constant multiplier can be adjusted.

Worked example

Differentiate (x² + 4)⁴, then find ∫6x(x² + 4)³ dx.

\begin{gathered}\frac{d}{dx}(x^2+4)^4\\=4(x^2+4)^3\times2x\\=8x(x^2+4)^3\end{gathered}

The requested integrand is $6/8=3/4$ of that derivative. Multiply the original function by the same constant.

\begin{gathered}\int6x(x^2+4)^3\,dx\\=\frac34(x^2+4)^4+c\end{gathered}

Match the variable factor, not just the bracket

Your turnChoose the method

Find both integrals: $\int3x(2x^2+1)^4\,dx$ and $\int(x^2+1)^2\,dx$ .

Show worked answer

The first has the inner derivative factor: differentiating $(2x^2+1)^5$ produces $20x(2x^2+1)^4$ . The second needs expansion.

Answer

\begin{gathered}\frac3{20}(2x^2+1)^5+c\\\frac{x^5}{5}+\frac{2x^3}{3}+x+c\end{gathered}

These are separate indefinite integrals; their arbitrary constants need not have the same value.

Your turnPractice

Differentiate $(x^2+3)^{-1}$ . Hence find $\int4x/(x^2+3)^2\,dx$ .

Show worked answer

\begin{gathered}\frac{d}{dx}(x^2+3)^{-1}\\=-2x(x^2+3)^{-2}\end{gathered}

The requested integrand is −2 times this derivative.

Answer

-\frac2{x^2+3}+c