Solve ∫ {left(-frac{7/5bx^3right)}^2}{left(-2x^2right)^3 wrt x

$$\int{ \frac{ { \left(- \frac{ 7 }{ 5 } b { x }^{ 3 } \right) }^{ 2 } }{ { \left(-2 { x }^{ 2 } \right) }^{ 3 } } }d x$$

Evaluate

$$-\frac{49xb^{2}}{200}+С$$

Solution Steps

Expand \left(-\frac{7}{5}bx^{3}\right)^{2}.

$$\int \frac{\left(-\frac{7}{5}\right)^{2}b^{2}\left(x^{3}\right)^{2}}{\left(-2x^{2}\right)^{3}}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\int \frac{\left(-\frac{7}{5}\right)^{2}b^{2}x^{6}}{\left(-2x^{2}\right)^{3}}\mathrm{d}x$$

Calculate -\frac{7}{5} to the power of 2 and get \frac{49}{25}.

$$\int \frac{\frac{49}{25}b^{2}x^{6}}{\left(-2x^{2}\right)^{3}}\mathrm{d}x$$

Expand \left(-2x^{2}\right)^{3}.

$$\int \frac{\frac{49}{25}b^{2}x^{6}}{\left(-2\right)^{3}\left(x^{2}\right)^{3}}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\int \frac{\frac{49}{25}b^{2}x^{6}}{\left(-2\right)^{3}x^{6}}\mathrm{d}x$$

Calculate -2 to the power of 3 and get -8.

$$\int \frac{\frac{49}{25}b^{2}x^{6}}{-8x^{6}}\mathrm{d}x$$

Cancel out x^{6} in both numerator and denominator.

$$\int \frac{\frac{49}{25}b^{2}}{-8}\mathrm{d}x$$

Divide \frac{49}{25}b^{2} by -8 to get -\frac{49}{200}b^{2}.

$$\int -\frac{49}{200}b^{2}\mathrm{d}x$$

Find the integral of -\frac{49b^{2}}{200} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(-\frac{49b^{2}}{200}\right)x$$

Simplify.

$$-\frac{49b^{2}x}{200}$$

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.

$$-\frac{49b^{2}x}{200}+С$$

Differentiate w.r.t. x

$$-\frac{49b^{2}}{200}$$