Solve ∫ (a^-7)^-2(a^5)^-1

$$\int ( a ^ { - 7 } ) ^ { - 2 } ( a ^ { 5 } ) ^ { - 1 }$$

Evaluate

$$xa^{9}+С$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -7 and -2 to get 14.

$$\int a^{14}\left(a^{5}\right)^{-1}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 5 and -1 to get -5.

$$\int a^{14}a^{-5}\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 14 and -5 to get 9.

$$\int a^{9}\mathrm{d}x$$

Find the integral of a^{9} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$a^{9}x$$

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.

$$a^{9}x+С$$

Differentiate w.r.t. x

$$a^{9}$$