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Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int 2 x \sec ^{3}\left(x^{2}+3\right) \tan \left(x^{2}+3\right) d x$

Solution:

$\sec ^{3}\left(x^{2}+3\right)$ can be written as $\sec ^{2}\left(x^{2}+3\right) \cdot \sec \left(x^{2}+3\right)$

Now the question becomes

$\Rightarrow \int 2 \mathrm{x} \cdot \sec ^{2}\left(\mathrm{x}^{2}+3\right) \sec \left(\mathrm{x}^{2}+3\right) \tan \left(\mathrm{x}^{2}+3\right) \mathrm{dx}$

Assume $\sec \left(x^{2}+3\right)=t$

$d\left(\sec \left(x^{2}+3\right)\right)=d t$

$2 x \sec \left(x^{2}+3\right) \tan \left(x^{2}+3\right) d x=d t$

Substituting $t$ and dt in the given equation

$\Rightarrow \int \mathrm{t}^{2} \mathrm{dt}$

$\Rightarrow \frac{\mathrm{t}^{3}}{3}+\mathrm{c}$

$\Rightarrow \frac{1}{3}\left(\sec \left(x^{2}+3\right)^{3}\right)+c$