Joni Kenny

2021-08-18

Find $\int 2{x}^{3}\mathrm{cos}\left({x}^{2}\right)dx$

Brittany Patton

Skilled2021-08-19Added 100 answers

We have to find the following integral

$\int 2{x}^{3}\mathrm{cos}\left({x}^{2}\right)dx$

Substitute$u={x}^{2}$ we have

$du=2xdx$

Thus we have

int$2{x}^{3}\mathrm{cos}\left({x}^{2}\right)dx=\int u\mathrm{cos}\left(u\right)du$

=$u\mathrm{sin}\left(u\right)-\int \mathrm{sin}\left(u\right)du$ [Using integration by parts]

=${x}^{2}\mathrm{sin}\left({x}^{2}\right)-(-\mathrm{cos}\left({x}^{2}\right))+C$

$\int 2{x}^{3}\mathrm{cos}\left({x}^{2}\right)dx={x}^{2}\mathrm{sin}\left({x}^{2}\right)-(-\mathrm{cos}\left({x}^{2}\right))+C$

Substitute

Thus we have

int

=

=

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