The value of lambda for which int frac{4x^3 + | vedclass

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(B) Let $I = \int \frac{4x^3 + \lambda 4^x}{4^x + x^4} \, dx$. Given that $I = \log(4^x + x^4) + c$. Let $f(x) = 4^x + x^4$. Then,the derivative $f'(x

Vedclass · Accepted Answer

$\ln 4$

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(B) Let $I = \int \frac{4x^3 + \lambda 4^x}{4^x + x^4} \, dx$. Given that $I = \log(4^x + x^4) + c$. Let $f(x) = 4^x + x^4$. Then,the derivative $f'(x) = \frac{d}{dx}(4^x + x^4) = 4^x \ln 4 + 4x^3$. We know that $\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + c$. Comparing the given integral $\int \frac{4x^3 + \lambda 4^x}{4^x + x^4} \, dx$ with $\int \frac{f'(x)}{f(x)} \, dx$,we have: $4x^3 + \lambda 4^x = 4x^3 + 4^x \ln 4$. Equating the coefficients of $4^x$,we get $\lambda = \ln 4$.

The value of $\lambda$ for which $\int \frac{4x^3 + \lambda 4^x}{4^x + x^4} \, dx = \log (4^x + x^4) + c$ is

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