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0-sin-2-x-sin-1-t-dt-0-cos-2-x-cos-1-t-dt-




Question Number 5321 by qw last updated on 07/May/16
 ∫_( 0) ^(sin^2 x)  sin^(−1) (√t) dt +∫_( 0) ^(cos^2 x) cos^(−1) (√t) dt =
$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{sin}^{\mathrm{2}} {x}} {\int}}\:\mathrm{sin}^{−\mathrm{1}} \sqrt{{t}}\:{dt}\:+\underset{\:\mathrm{0}} {\overset{\mathrm{cos}^{\mathrm{2}} {x}} {\int}}\mathrm{cos}^{−\mathrm{1}} \sqrt{{t}}\:{dt}\:= \\ $$
Commented by Yozzii last updated on 08/May/16
I=∫_0 ^(sin^2 x) sin^(−1) (√t) dt+∫_0 ^(cos^2 x) cos^(−1) (√t) dt  I=−(1/2)cos2xsin^(−1) ∣sinx∣+((1+cos2x)/2)cos^(−1) ∣cosx∣+(1/2)sin^(−1) ∣sin((π/2)−x)∣  For 0<x<(π/2),  I=∫_0 ^(sin^2 x) sin^(−1) (√t) dt+∫_0 ^(cos^2 x) cos^(−1) (√t) dt=(π/4).
$${I}=\int_{\mathrm{0}} ^{{sin}^{\mathrm{2}} {x}} {sin}^{−\mathrm{1}} \sqrt{{t}}\:{dt}+\int_{\mathrm{0}} ^{{cos}^{\mathrm{2}} {x}} {cos}^{−\mathrm{1}} \sqrt{{t}}\:{dt} \\ $$$${I}=−\frac{\mathrm{1}}{\mathrm{2}}{cos}\mathrm{2}{xsin}^{−\mathrm{1}} \mid{sinx}\mid+\frac{\mathrm{1}+{cos}\mathrm{2}{x}}{\mathrm{2}}{cos}^{−\mathrm{1}} \mid{cosx}\mid+\frac{\mathrm{1}}{\mathrm{2}}{sin}^{−\mathrm{1}} \mid{sin}\left(\frac{\pi}{\mathrm{2}}−{x}\right)\mid \\ $$$${For}\:\mathrm{0}<{x}<\frac{\pi}{\mathrm{2}}, \\ $$$${I}=\int_{\mathrm{0}} ^{{sin}^{\mathrm{2}} {x}} {sin}^{−\mathrm{1}} \sqrt{{t}}\:{dt}+\int_{\mathrm{0}} ^{{cos}^{\mathrm{2}} {x}} {cos}^{−\mathrm{1}} \sqrt{{t}}\:{dt}=\frac{\pi}{\mathrm{4}}. \\ $$

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