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  4. The value of the integral displaystyle ∫ 01 4 t3 (1 + t)8 + 8 t4 (1 + t)7 dt is

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Q. The value of the integral $\displaystyle \int _{0}^{1} \left\{4 t^{3} \left(1 + t\right)^{8} + 8 t^{4} \left(1 + t\right)^{7}\right\}dt$ is

NTA AbhyasNTA Abhyas 2020Integrals

Solution:

Note that $\frac{d}{d t}\left\{t^{4} \left(1 + t\right)^{8}\right\}=4t^{3}\left(1 + t\right)^{8}+8t^{4}\left(1 + t\right)^{7}$
Hence, the required integral is
$\left.t^{4}(1+t)^{8}\right|_{0} ^{1}=2^{8}$