Calculus

Integration of Functions

Change of Variable

Page 1
Problems 1-2
Page 2
Problems 3-5

Let \(F\left( x \right)\) be an indefinite integral or antiderivative of \(f\left( x \right).\) Then

\[
{\int {f\left( x \right)dx} }
= {\int {f\left( {g\left( u \right)} \right)g’\left( u \right)du} }
= {F\left( u \right) }
= {F\left( {{g^{ – 1}}\left( x \right)} \right),}
\]

Solved Problems

Click on problem description to see solution.

 Example 1

Calculate the integral \(\int {\large\frac{{dx}}{{\sqrt {{a^2} – {x^2}} }}\normalsize} .\)

 Example 2

Find the integral \(\int {{\large\frac{{x + 1}}{{{x^2} + 2x – 5}}\normalsize} dx}. \)

 Example 3

Calculate the integral \(\int {{2^x}{e^x}dx} .\)

 Example 4

Calculate the integral \(\int {\cot \left( {3x + 5} \right)dx}.\)

 Example 5

Find the integral \(\int {{\large\frac{{\sin 2x}}{{\sqrt {1 + {{\cos }^2}x} }}\normalsize} dx}.\)

Example 1.

Calculate the integral \(\int {\large\frac{{dx}}{{\sqrt {{a^2} – {x^2}} }}\normalsize} .\)

Solution.

Let \(u = \large\frac{x}{a}\normalsize.\) Then \(x = au,\) \(dx = adu.\) Hence, the integral is

\[\require{cancel}
{\int {\frac{{dx}}{{\sqrt {{a^2} – {x^2}} }}} }
= {\int {\frac{{adu}}{{\sqrt {{a^2} – {{\left( {au} \right)}^2}} }}} }
= {\int {\frac{{adu}}{{\sqrt {{a^2}\left( {1 – {u^2}} \right)} }}} }
= {\int {\frac{{\cancel{a}du}}{{\cancel{a}\sqrt {1 – {u^2}} }}} }
= {\int {\frac{{du}}{{\sqrt {1 – {u^2}} }}} }
= {\arcsin u + C }
= {\arcsin \frac{x}{a} + C.}
\]

Example 2.

Find the integral \(\int {{\large\frac{{x + 1}}{{{x^2} + 2x – 5}}\normalsize} dx}. \)

Solution.

We make the substitution \(u = {x^2} + 2x – 5.\) Then \(du = 2xdx + 2dx \) \(= 2\left( {x + 1} \right)dx\) or \(\left( {x + 1} \right)dx = {\large\frac{{du}}{2}\normalsize}.\) The integral is easy to calculate with the new variable:

\[
{\int {\frac{{x + 1}}{{{x^2} + 2x – 5}}dx} }
= {\int {\frac{{\frac{{du}}{2}}}{u}} }
= {\frac{1}{2}\int {\frac{{du}}{u}} }
= {\frac{1}{2}\ln \left| u \right| + C }
= {\frac{1}{2}\ln \left| {{x^2} + 2x – 5} \right| }+{ C.}
\]
Page 1
Problems 1-2
Page 2
Problems 3-5