Please,help me!It is integral with substitution

1.	Please,help me!It is integral with substitution
Answer» Question : Integrate $1) \int \dfrac{cosx \: dx}{4 + sin {}^{2}x}$ $2) \int \dfrac{dx}{x(1 + ln {}^{2} x)}$ Theory: Integration is the inverse process of Differentiation. •Methods of integration Integration by substitution : The GIVEN integral $\int f(x)dx$ can be transformed into another by changing the independent VARIABLE x to t by SUBSTITUTING x = g(t) •Formula's used : $1) \int \frac{1}{x {}^{2} + a {}^{2} } = \frac{1}{a} \tan {}^{ - 1} ( \frac{x}{a} )$ $2) \int \frac{dx}{1 + x {}^{2} } = \tan {}^{ - 1} (x)$ Solution : $1) I= \int \bf \dfrac{cosx \: dx}{4 + sin {}^{2} x}$ Let sinx = t $\frac{dy}{dx} = cosx\implies \frac{<klux>DT</klux>}{cosx} = dx$ $I= \int \frac{ \cancel cosx}{4 + t {}^{2} } \times \frac{dt}{ \cancel cosx}$ $I= \int \frac{dt}{4+ t {}^{2} }$ $I= \int \frac{dt}{2 {}^{2} + t {}^{2} }$ use formula : $\int\:\frac{dx}{x{}^{2}+a{}^{2}}$ $I = \frac{1}{2} \tan {}^{ - 1} ( \frac{t}{2} ) + c$ Put t = sinx $I= \dfrac{1}{2} \tan( \frac{ \sin x}{2} ) + c$ __________________________ $2) I=\int \bf \dfrac{dx}{x(1 +ln {}^{2} x) }$ Let lnx = t ⇒xdt = dt $\int \bf \frac{dx}{x(1 + ln {}^{2}x) } = \int \frac{ \cancel xdt}{ \cancel x(1 + t {}^{2} )}$ $I = \int \frac{dt}{1 + t {}^{2} }$ $I= \tan {}^{ - 1} (t) + c$ put t = ln x $I= \tan {}^{ - 1} (lnx) + c$

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