Find a function f such that f‴(x)=11+sin2x. (This problem is supposed to be easy; don't...

Jaya Legge

Answered question

2021-06-25

Find a function f such that $f{}^{\u2034}\left(x\right)=\frac{1}{\sqrt{1+{\mathrm{sin}}^{2}x}}$. (This problem is supposed to be easy; don't misinterpret the word "find".)

Answer & Explanation

Willie

Skilled2021-06-26Added 95 answers

If ${f}^{\prime}\left(x\right)=g\left(x\right)\text{}then\text{}f\left(x\right)=\int g\left(x\right)dx$ Thus, $\left(f{}^{\u2033}\left(x\right)\right)}^{\prime}=\frac{1}{\sqrt{1+{\mathrm{sin}}^{2}\left(x\right)}$ $f{}^{\u2033}\left(x\right){=}^{\left(1\right)}\int \frac{1}{\sqrt{1+{\mathrm{sin}}^{2}+\left(x\right)}}dx$ The above equation, can be written as: ${\left({f}^{\prime}\left(x\right)\right)}^{\prime}=\int \frac{1}{\sqrt{1+{\mathrm{sin}}^{2}\left(x\right)}}dx$ So, ${f}^{\prime}\left(x\right){=}^{\left(1\right)}\int \int \frac{1}{\sqrt{1+{\mathrm{sin}}^{2}\left(x\right)}}dxdx$ Now, from (1) we get $f\left(x\right)=\int \int \int \frac{1}{\sqrt{1+{\mathrm{sin}}^{2}\left(x\right)}}dxdxdx$