integral of 2/3 x^{3/2}-9sin(x) — step-by-step solution

⚠Ⓘ 2/3 x^{3/2}-9sin(x) is read as (2/3)*x^(3/2)-9sin(x): use parentheses around the denominator to include the following factor in it, use ^(...) to group an exponent, use * to multiply

1) Before integrating, let's calculate:

\frac{2}{3} \cdot x^{\frac{3}{2}} - 9 \cdot \sin (x)

\frac{2 x^{\frac{3}{2}}}{3} - 9 \cdot \sin (x)

2) Let's integrate:

\int(\frac{2 x^{\frac{3}{2}}}{3} - 9 \cdot \sin (x))dx

For the math expression:

- 9 \cdot \sin (x)

, we know the derivative formula:

\frac{d}{d x} (\cos (a)) = - \sin (a)

, so we try differentiating with cos

3) Let's differentiate:

\frac{d}{d x} (\cos (x))

- \sin (x)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

9 \cdot \cos (x) + \int(\frac{2 x^{\frac{3}{2}}}{3})dx

We integrate

\frac{2 x^{\frac{3}{2}}}{3}

by applying:

\int(x^{n})dx = \frac{x^{(n + 1)}}{n + 1}

9 \cdot \cos (x) + \frac{2}{3} \cdot \frac{x^{(\frac{3}{2} + 1)}}{\frac{3}{2} + 1}

ℹ

\frac{2}{3} \cdot \frac{x^{(\frac{3}{2} + 1)}}{\frac{3}{2} + 1}

\frac{2 x^{(\frac{3}{2} + 1)}}{3 \cdot (\frac{3}{2} + 1)}

‖

\frac{3}{2} + 1

\frac{3 + 2}{2}

9 \cdot \cos (x) + \frac{2 x^{(\frac{3 + 2}{2})}}{3 \cdot \frac{3 + 2}{2}}

ℹ

3 + 2

5

‖

3 \cdot \frac{3 + 2}{2}

\frac{3 \cdot (3 + 2)}{2}

9 \cdot \cos (x) + \frac{2 x^{\frac{5}{2}}}{\frac{3 \cdot 5}{2}}

9 \cdot \cos (x) + \frac{2 x^{\frac{5}{2}}}{\frac{15}{2}}

9 \cdot \cos (x) + 2 x^{\frac{5}{2}} \cdot \frac{2}{15}

9 \cdot \cos (x) + \frac{4}{15} \cdot x^{\frac{5}{2}}

9 \cdot \cos (x) + \frac{4 x^{\frac{5}{2}}}{15}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{4 x^{\frac{5}{2}}}{15} + 9 \cdot \cos (x) + C