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Differentiate w.r.t. x
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\int x^{3}arcsen\mathrm{d}x
To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.
arcsen\int x^{3}\mathrm{d}x
Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.
arcsen\times \frac{x^{4}}{4}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.
\frac{arecsnx^{4}}{4}
Simplify.
\frac{arecsnx^{4}}{4}+С
If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.