Evaluate the Integral integral of 1/(5x^3) with respect to x

Problem

(∫_^)(1/(5*x3)*d(x))

Solution

1. Rewrite the integrand by moving the constant factor outside the integral and expressing the variable with a negative exponent.

(∫_^)(1/(5*x3)*d(x))=1/5*(∫_^)(x(−3)*d(x))

2. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1)+C for n≠−1

1/5*(∫_^)(x(−3)*d(x))=1/5⋅(x(−3+1))/(−3+1)+C

3. Simplify the expression by performing the arithmetic in the exponent and the denominator.

1/5⋅(x(−2))/(−2)+C=−1/10*x(−2)+C

4. Convert the negative exponent back into a fraction for the final form.

−1/10*x(−2)+C=−1/(10*x2)+C

Final Answer

(∫_^)(1/(5*x3)*d(x))=−1/(10*x2)+C

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