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The solution to your problem is this : \begin{align} R(T) & =\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} \lambda^{-5}f(\lambda T)\mathrm{d}\lambda=\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} T^{4}\dfrac{f(\lambda T)}{(\lambda T)^{5}}\mathrm{d}(\lambda T)\qquad \Longrightarrow \nonumber\\ R(T) & =\frac c4 \underbrace{\left(\:\:\int\limits_{\mu=0}^{\mu=\infty} \dfrac{f(\mu)}{\mu^{5}}\mathrm{d}\mu\right)}_{A=\text{constant}}T^{4}= \underbrace{\left(\frac c4 A\right)}_{\sigma}\, T^{4}=\sigma\, T^{4} \tag{01} \end{align}

But, sincerely, trying to find this directly you are missing important facts about the "before Planck" adventure of the blackbody radiation theory. For example, you must try to find from Wien's Law (your first equation) why if you know the function $\;\rho(\lambda,T_{1})\;$ for a given temperature $\;T_{1}\;$ then you know it for any temperature $\;T\;$ or that $\;\lambda_{\rm max}\cdot T=b=\rm constant\;$ (Wien's Displacement Law), see Emilio Pisanty answer therein : Showing Wien's Displacement Law from Wien's Law.

The solution to your problem⁽¹⁾ is this : \begin{align} R(T) & =\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} \lambda^{-5}f(\lambda T)\mathrm{d}\lambda=\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} T^{4}\dfrac{f(\lambda T)}{(\lambda T)^{5}}\mathrm{d}(\lambda T)\qquad \Longrightarrow \nonumber\\ R(T) & =\frac c4 \underbrace{\left(\:\:\int\limits_{\mu=0}^{\mu=\infty} \dfrac{f(\mu)}{\mu^{5}}\mathrm{d}\mu\right)}_{A=\text{constant}}T^{4}= \underbrace{\left(\frac c4 A\right)}_{\sigma}\, T^{4}=\sigma\, T^{4} \tag{01} \end{align}

But, sincerely, trying to find this directly you are missing important facts about the "before Planck" adventure of the blackbody radiation theory. For example, you must try to find from Wien's Law (your first equation) why if you know the function $\;\rho(\lambda,T_{1})\;$ for a given temperature $\;T_{1}\;$ then you know it for any temperature $\;T\;$ or that $\;\lambda_{\rm max}\cdot T=b=\rm constant\;$ (Wien's Displacement Law), see Emilio Pisanty answer therein : Showing Wien's Displacement Law from Wien's Law.

The solution to your problem is this : \begin{align} R(T) & =\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} \lambda^{-5}f(\lambda T)\mathrm{d}\lambda=\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} T^{4}\dfrac{f(\lambda T)}{(\lambda T)^{5}}\mathrm{d}(\lambda T)\qquad \Longrightarrow \nonumber\\ R(T) & =\frac c4 \underbrace{\left(\:\:\int\limits_{\mu=0}^{\mu=\infty} \dfrac{f(\mu)}{\mu^{5}}\mathrm{d}\mu\right)}_{A=\text{constant}}T^{4}= \underbrace{\left(\frac c4 A\right)}_{\sigma}\, T^{4}=\sigma\, T^{4} \tag{01} \end{align}

But, sincerely, trying to find this directly you are missing important facts about the "before Planck" adventure of the blackbody radiation theory. For example, you must try to find from Wien's law (your first equation) why if you know the function $\;\rho(\lambda,T_{1})\;$ for a given temperature $\;T_{1}\;$ then you know it for any temperature $\;T\;$ or that $\;\lambda_{\rm max}\cdot T=b=\rm constant\;$ (Wien's displacement law).

The solution to your problem is this : \begin{align} R(T) & =\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} \lambda^{-5}f(\lambda T)\mathrm{d}\lambda=\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} T^{4}\dfrac{f(\lambda T)}{(\lambda T)^{5}}\mathrm{d}(\lambda T)\qquad \Longrightarrow \nonumber\\ R(T) & =\frac c4 \underbrace{\left(\:\:\int\limits_{\mu=0}^{\mu=\infty} \dfrac{f(\mu)}{\mu^{5}}\mathrm{d}\mu\right)}_{A=\text{constant}}T^{4}= \underbrace{\left(\frac c4 A\right)}_{\sigma}\, T^{4}=\sigma\, T^{4} \tag{01} \end{align}

But, sincerely, trying to find this directly you are missing important facts about the "before Planck" adventure of the blackbody radiation theory. For example, you must try to find from Wien's Law (your first equation) why if you know the function $\;\rho(\lambda,T_{1})\;$ for a given temperature $\;T_{1}\;$ then you know it for any temperature $\;T\;$ or that $\;\lambda_{\rm max}\cdot T=b=\rm constant\;$ (Wien's Displacement Law), see Emilio Pisanty answer therein : Showing Wien's Displacement Law from Wien's Law.

The solution to your problem is this : \begin{align} R(T) & =\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} \lambda^{-5}f(\lambda T)\mathrm{d}\lambda=\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} T^{4}\dfrac{f(\lambda T)}{(\lambda T)^{5}}\mathrm{d}(\lambda T)\qquad \Longrightarrow \nonumber\\ R(T) & =\frac c4 \underbrace{\left(\:\:\int\limits_{\mu=0}^{\mu=\infty} \dfrac{f(\mu)}{\mu^{5}}\mathrm{d}\mu\right)}_{A=\text{constant}}T^{4}= \underbrace{\left(\frac c4 A\right)}_{\sigma}\, T^{4}=\sigma\, T^{4} \tag{01} \end{align}

But, sincerely, trying to find this you are missing important facts about the "before Planck" adventure of the blackbody radiation theory.

The solution to your problem is this : \begin{align} R(T) & =\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} \lambda^{-5}f(\lambda T)\mathrm{d}\lambda=\frac c4 \int\limits_{\lambda=0}^{\lambda=\infty} T^{4}\dfrac{f(\lambda T)}{(\lambda T)^{5}}\mathrm{d}(\lambda T)\qquad \Longrightarrow \nonumber\\ R(T) & =\frac c4 \underbrace{\left(\:\:\int\limits_{\mu=0}^{\mu=\infty} \dfrac{f(\mu)}{\mu^{5}}\mathrm{d}\mu\right)}_{A=\text{constant}}T^{4}= \underbrace{\left(\frac c4 A\right)}_{\sigma}\, T^{4}=\sigma\, T^{4} \tag{01} \end{align}

But, sincerely, trying to find this directly you are missing important facts about the "before Planck" adventure of the blackbody radiation theory. For example, you must try to find from Wien's law (your first equation) why if you know the function $\;\rho(\lambda,T_{1})\;$ for a given temperature $\;T_{1}\;$ then you know it for any temperature $\;T\;$ or that $\;\lambda_{\rm max}\cdot T=b=\rm constant\;$ (Wien's displacement law).