Examples On Integration By Substitution Set-5

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Example - 19

Evaluate the following integrals

 

(a) \(\begin{align} \int {\frac{1}{{\sqrt {4x - {x^2} - 3} }}dx} \end{align}\) (b)\(\begin{align} \int {\frac{1}{{{x^2} + 2x + 2}}dx} \end{align}\)
(c) \(\begin{align} \int {\frac{1}{{(x + 2)\sqrt {{x^2} + 4x + 3} }}dx} \end{align}\) (d) \(\begin{align} \int {\frac{1}{{{x^2} + 6x + 8}}\,} dx \end{align}\)
(e) \(\begin{align} \int {\frac{1}{{\sqrt {4{x^2} + 12x + 10} }}dx}  \end{align}\) (f)\(\begin{align} \int {\frac{1}{{\sqrt {25{x^2} + 20x + 3} }}dx} \end{align}\)

 

Solution: If you observe the forms of the expressions to be integrated carefully, you will realise that each part corresponds to a part in the previous example. Hence we’ll use the results obtained in the previous part directly:

(a) \[I = \int {\frac{1}{{\sqrt {4x - {x^2} - 3} }}dx} \]

\[\begin{align}&\,\,\,\, = \int {\frac{1}{{\sqrt {1 - {{(x - 2)}^2}} }}dx} \\&\,\,\,\, = {\sin ^{ - 1}}(x - 2) + C\end{align}\]

(b) \[I = \int {\frac{1}{{{x^2} + 2x + 2}}dx} \]

\[\begin{align}&\,\,\,\, = \int {\frac{1}{{{{(x + 1)}^2} + 1}}dx} \\&\,\,\,\, = {\tan ^{ - 1}}(x + 1) + C\end{align}\]

(c) \[I = \int {\frac{1}{{(x + 2)\sqrt {{x^2} + 4x + 3} }}} dx\]

\[\begin{align}&\,\,\,\, = \int {\frac{1}{{(x + 2)\sqrt {{{(x + 2)}^2} - 1} }}} dx\\&\,\,\,\, = {\sec ^{ - 1}}(x + 2) + C\end{align}\]

(d) \[I = \int {\frac{1}{{{x^2} + 6x + 8}}dx} \]

\[\begin{align}&\,\,\,\, = \int {\frac{1}{{{{(x + 3)}^2} - 1}}dx} \\&\,\,\,\, = \frac{1}{2}\ln \left| {\frac{{x + 2}}{{x + 4}}} \right| + C\end{align}\]

(e) \[I = \int {\frac{1}{{\sqrt {4{x^2} + 12x + 10} }}dx} \]

\[\begin{align}&\qquad\qquad \qquad\;= \int {\frac{1}{{\sqrt {{{(2x + 3)}^2} + 1} }}dx} \\&\qquad\qquad\qquad\;= \frac{1}{2}\ln \left| {(2x + 3) + \sqrt {4{x^2} + 12x + 10} } \right| + C\end{align}\]

(f) \[I = \int {\frac{1}{{\sqrt {25{x^2} + 20x + 3} }}dx} \]

\[\begin{align}&\qquad\qquad\qquad\;\;= \int {\frac{1}{{\sqrt {{{(5x + 2)}^2} - 1} }}dx} \\& \qquad\qquad\qquad\;\; = \frac{1}{5}\ln \left| {(5x + 2) + \sqrt {25{x^2} + 20x + 3} } \right| + C\end{align}\]

Download SOLVED Practice Questions of Examples On Integration By Substitution Set-5 for FREE
Indefinite Integration
grade 11 | Questions Set 1
Indefinite Integration
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Download SOLVED Practice Questions of Examples On Integration By Substitution Set-5 for FREE
Indefinite Integration
grade 11 | Questions Set 1
Indefinite Integration
grade 11 | Answers Set 1
Indefinite Integration
grade 11 | Questions Set 2
Indefinite Integration
grade 11 | Answers Set 2
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