Mathematics

# Integrate $\int {({{\sin }^{ - 1}}} x{)^2}dx$

##### SOLUTION
$\begin{array}{l} I=\int { { { \left( { { { \sin }^{ -1 } }x } \right) }^{ 2 } }dx } & \\ I=\int { \left( { { { \sin }^{ -1 } }x } \right) .\left( { { { \sin }^{ -1 } }x } \right) dx } & \left[ \begin{array}{l} Put\, \, { \sin ^{ -1 } }=t \\ x=\sin t \\ dx=\cos tdt \end{array} \right] \\ =\int { { t^{ 2 } }.\cos tdt } & \end{array}$
Using $ILATE$ formulae
$\begin{array}{l} ={ t^{ 2 } }\sin t-\int { 2t\sin tdt } \\ ={ t^{ 2 } }\sin t-2\int { t\sin tdt } \to Apply\, \, ILATE\, \, formulae \\ ={ t^{ 2 } }\sin t-2\left[ { -t\cos t+\int { \cos tdt } } \right] \\ ={ t^{ 2 } }\sin t+2t\cos t-2\sin t+c \\ ={ \left( { { { \sin }^{ -1 } }x } \right) ^{ 2 } }\sin .{ \sin ^{ -1 } }x+2{ \sin ^{ -1 } }x\sqrt { 1-{ x^{ 2 } } } -2\sin .{ \sin ^{ -1 } }x+c \\ ={ \left( { { { \sin }^{ -1 } }x } \right) ^{ 2 } }.x+2{ \sin ^{ -1 } }x\sqrt { 1-{ x^{ 2 } } } -2x+c \end{array}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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