Calculus Examples

$\int \frac{t^{2} - π \sqrt{t} + \sqrt{t} \csc (t)}{\sqrt{t}} d t$

Step 1

Simplify the expression.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\int \frac{t^{2} - π t^{\frac{1}{2}} + \sqrt{t} \csc (t)}{\sqrt{t}} d t$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\int \frac{t^{2} - π t^{\frac{1}{2}} + t^{\frac{1}{2}} \csc (t)}{\sqrt{t}} d t$

Step 1.2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\int \frac{t^{2} - π t^{\frac{1}{2}} + t^{\frac{1}{2}} \csc (t)}{t^{\frac{1}{2}}} d t$

Step 1.3

Simplify.

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Step 1.3.1

Factor $t^{\frac{1}{2}}$ out of $t^{2} - π t^{\frac{1}{2}} + t^{\frac{1}{2}} \csc (t)$ .

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Step 1.3.1.1

Factor $t^{\frac{1}{2}}$ out of $t^{2}$ .

$\int \frac{t^{\frac{1}{2}} t^{\frac{3}{2}} - π t^{\frac{1}{2}} + t^{\frac{1}{2}} \csc (t)}{t^{\frac{1}{2}}} d t$

Step 1.3.1.2

Factor $t^{\frac{1}{2}}$ out of $- π t^{\frac{1}{2}}$ .

$\int \frac{t^{\frac{1}{2}} t^{\frac{3}{2}} + t^{\frac{1}{2}} (- π) + t^{\frac{1}{2}} \csc (t)}{t^{\frac{1}{2}}} d t$

Step 1.3.1.3

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} \csc (t)$ .

$\int \frac{t^{\frac{1}{2}} t^{\frac{3}{2}} + t^{\frac{1}{2}} (- π) + t^{\frac{1}{2}} (\csc (t))}{t^{\frac{1}{2}}} d t$

Step 1.3.1.4

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} t^{\frac{3}{2}} + t^{\frac{1}{2}} (- π)$ .

$\int \frac{t^{\frac{1}{2}} (t^{\frac{3}{2}} - π) + t^{\frac{1}{2}} (\csc (t))}{t^{\frac{1}{2}}} d t$

Step 1.3.1.5

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} (t^{\frac{3}{2}} - π) + t^{\frac{1}{2}} (\csc (t))$ .

$\int \frac{t^{\frac{1}{2}} (t^{\frac{3}{2}} - π + \csc (t))}{t^{\frac{1}{2}}} d t$

Step 1.3.2

Cancel the common factor.

$\int \frac{t^{\frac{1}{2}} (t^{\frac{3}{2}} - π + \csc (t))}{t^{\frac{1}{2}}} d t$

Step 1.3.3

Divide $t^{\frac{3}{2}} - π + \csc (t)$ by $1$ .

$\int t^{\frac{3}{2}} - π + \csc (t) d t$

Step 2

Split the single integral into multiple integrals.

$\int t^{\frac{3}{2}} d t + \int - π d t + \int \csc (t) d t$

Step 3

By the Power Rule, the integral of $t^{\frac{3}{2}}$ with respect to $t$ is $\frac{2}{5} t^{\frac{5}{2}}$ .

$\frac{2}{5} t^{\frac{5}{2}} + C + \int - π d t + \int \csc (t) d t$

Step 4

Apply the constant rule.

$\frac{2}{5} t^{\frac{5}{2}} + C - π t + C + \int \csc (t) d t$

Step 5

The integral of $\csc (t)$ with respect to $t$ is $\ln (| \csc (t) - \cot (t) |)$ .

$\frac{2}{5} t^{\frac{5}{2}} + C - π t + C + \ln (| \csc (t) - \cot (t) |) + C$

Step 6

Simplify.

$\frac{2}{5} t^{\frac{5}{2}} - π t + \ln (| \csc (t) - \cot (t) |) + C$