Calculus

Integration of Functions

Integration of Functions Logo

Trigonometric and Hyperbolic Substitutions

  • In this section we consider integrals of the form \({\large\int\normalsize} {R\left( {x,\sqrt {a{x^2} + bx + c} } \right)dx} ,\) where \(R\) is a rational function of \(x\) and the radical \({\sqrt {a{x^2} + bx + c} }.\)

    To calculate such an integral, we need first to complete the square in the quadratic expression:

    \[
    {a{x^2} + bx + c }
    = {a\left( {{{\left( {x + \frac{b}{{2a}}} \right)}^2} }\right.}+{\left.{ \left( {c – \frac{{{b^2}}}{{4a}}} \right)} \right).}
    \]

    Making the substitution \(u = x + \large\frac{b}{{2a}}\normalsize,\) \(du = dx,\) we can obtain one of the three possible integrals, depending on the values of the coefficients \(a, b\) and \(c:\)

    1. \({\large\int\normalsize} {R\left( {u,\sqrt {{r^2} – {u^2}} } \right)du} \)
    2. \({\large\int\normalsize} {R\left( {u,\sqrt {{r^2} + {u^2}} } \right)du} \)
    3. \({\large\int\normalsize} {R\left( {u,\sqrt {{u^2} – {r^2}} } \right)du} \)

    Then we use the following trigonometric or hyperbolic substitutions to simplify the integrals:

    1. Integrals of the form \({\large\int\normalsize} {R\left( {u,\sqrt {{r^2} – {u^2}} } \right)du} \)

    Trigonometric substitution:

    \[
    {u = r\sin t,\;\;}\kern-0.3pt
    {du = r\cos tdt,\;\;}\kern-0.3pt
    {\sqrt {{r^2} – {u^2}} = r\cos t,\;\;}\kern-0.3pt
    {t = \arcsin \left( {\frac{u}{r}} \right).}
    \]

    2. Integrals of the form \({\large\int\normalsize} {R\left( {u,\sqrt {{r^2} + {u^2}} } \right)du} \)

    Trigonometric substitution:

    \[
    {u = r\tan t,\;\;}\kern-0.3pt
    {du = r\,{\sec ^2}tdt,\;\;}\kern-0.3pt
    {\sqrt {{r^2} + {u^2}} = r\sec t,\;\;}\kern-0.3pt
    {t = \arctan \left( {\frac{u}{r}} \right).}
    \]

    Hyperbolic substitution:

    \[
    {u = r\sinh t,\;\;}\kern-0.3pt
    {du = r\cosh tdt,\;\;}\kern-0.3pt
    {\sqrt {{r^2} + {u^2}} = r\cosh t,\;\;}\kern-0.3pt
    {t = \text{arcsinh} \left( {\frac{u}{r}} \right).}
    \]

    3. Integrals of the form \({\large\int\normalsize} {R\left( {u,\sqrt {{u^2} – {r^2}} } \right)du} \)

    Trigonometric substitution:

    \[
    {u = r\sec t,\;\;}\kern-0.3pt
    {du = r\tan t\sec tdt,\;\;}\kern-0.3pt
    {\sqrt {{u^2} – {r^2}} = r\tan t,\;\;}\kern-0.3pt
    {t = \arccos \left( {\frac{r}{u}} \right).}
    \]

    Hyperbolic substitution:

    \[
    {u = r\cosh t,\;\;}\kern-0.3pt
    {du = r\sinh tdt,\;\;}\kern-0.3pt
    {\sqrt {{u^2} – {r^2}} = r\sinh t,\;\;}\kern-0.3pt
    {t = \text{arccosh} \left( {\frac{u}{r}} \right).}
    \]

    Remarks.

    • Instead of the trigonometric substitutions in cases \(1, 2, 3\) you can use the substitutions \(x = r\cos t,\) \(x = r\cot t,\) \(x = r\csc t,\) respectively.
    • Using the formulas given above, we consider only positive values of the root. For example, in strict writing
      \[ {\sqrt {{r^2} – {u^2}} } = {\sqrt {{r^2} – {r^2}{{\cos }^2}t} } = {\sqrt {{r^2}{\sin^2}t} } = {\left| {r\sin t} \right|.} \]
      We suppose here that \(\left| {r\sin t} \right| \) \(= r\sin t.\)

  • Solved Problems

    Click a problem to see the solution.

    Example 1

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

    Example 2

    Evaluate the integral \({\large\int\normalsize} {\large\frac{{dx}}{{\sqrt {{{\left( {{a^2} + {x^2}} \right)}^3}} }}\normalsize}.\)

    Example 3

    Calculate the integral \({\large\int\normalsize} {\large\frac{{dx}}{{\sqrt {{{\left( {{x^2} – 8} \right)}^3}} }}\normalsize}.\)

    Example 4

    Calculate the integral \({\large\int\normalsize} {\large\frac{{dx}}{{\sqrt {\left( {x – a} \right)\left( {b – x} \right)} }}\normalsize}.\)

    Example 5

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

    Example 6

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

    Example 7

    Evaluate the integral \({\large\int\normalsize} {\sqrt {9{x^2} – 1} dx}.\)

    Example 8

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

    Example 1.

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

    Solution.

    We make the substitution:

    \[
    {x = a\sin t,\;\;}\kern-0.3pt
    {dx = a\cos tdt,\;\;}\kern0pt
    {t = \arcsin \frac{x}{a}.}
    \]

    Then

    \[
    {\int {\frac{{\sqrt {{a^2} – {x^2}} dx}}{{{x^2}}}} }
    = {\int {\frac{{\sqrt {{a^2} – {a^2}{{\sin }^2}t} }}{{{a^2}{{\sin }^2}t}}a\cos tdt} }
    = {\int {\frac{{a\cos t}}{{{a^2}{{\sin }^2}t}}a\cos tdt} }
    = {\int {{{\cot }^2}tdt} = \int {\left( {{{\csc }^2}t – 1} \right)dt} }
    = { – \cot t – t + C }={{ – \frac{{\sqrt {1 – {{\sin }^2}t} }}{{\sin t}} – t }+{ C }}
    = {{ – \frac{{\sqrt {1 – \frac{{{x^2}}}{{{a^2}}}} }}{{\frac{x}{a}}} – \arcsin \frac{x}{a} }+{ C }}
    = {{ – \frac{{\sqrt {{a^2} – {x^2}} }}{x} – \arcsin \frac{x}{a} }+{ C.}}
    \]

    To simplify the integral, we used here the trigonometric formula \({\cot ^2}t = {\csc ^2}t – 1.\)

    Example 2.

    Evaluate the integral \({\large\int\normalsize} {\large\frac{{dx}}{{\sqrt {{{\left( {{a^2} + {x^2}} \right)}^3}} }}\normalsize}.\)

    Solution.

    We make the hyperbolic substitution: \(x = a\sinh t,\) \(dx = a\cosh tdt.\) Using the hyperbolic identity \(1 + {\sinh ^2}t \) \(= {\cosh ^2}t,\) we can write:

    \[
    {\int {\frac{{dx}}{{\sqrt {{{\left( {{a^2} + {x^2}} \right)}^3}} }}} }
    = {\int {\frac{{a\cosh tdt}}{{\sqrt {{{\left( {{a^2} + {a^2}{{\sinh }^2}t} \right)}^3}} }}} }
    = {\int {\frac{{a\cosh tdt}}{{{a^3}{{\cosh }^3}t}}} }
    = {\frac{1}{{{a^2}}}\int {{{\text{sech}}^2}tdt} }
    = {\frac{1}{{{a^2}}}\tanh t + C }
    = {\frac{1}{{{a^2}}}\frac{{\sinh t}}{{\cosh t}} + C }
    = {\frac{1}{{{a^2}}}\frac{{\sinh t}}{{\sqrt {1 + {\sinh^2}t} }} + C }
    = {\frac{1}{{{a^2}}}\frac{{\frac{x}{a}}}{{\sqrt {1 + \frac{{{x^2}}}{{{a^2}}}} }} + C }
    = {\frac{1}{{{a^2}}}\frac{x}{{\sqrt {{a^2} + {x^2}} }} + C.}
    \]

    Page 1
    Problems 1-2
    Page 2
    Problems 3-8