Calculus Examples

$\int (\sqrt{t^{4} + 7 t}) (4 t^{3} + 7) d t$

Step 1

Simplify terms.

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Step 1.1

Factor $t$ out of $t^{4} + 7 t$ .

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Step 1.1.1

Factor $t$ out of $t^{4}$ .

$\frac{d}{d x} [\int \sqrt{t \cdot t^{3} + 7 t} (4 t^{3} + 7) d t]$

Step 1.1.2

Factor $t$ out of $7 t$ .

$\frac{d}{d x} [\int \sqrt{t \cdot t^{3} + t \cdot 7} (4 t^{3} + 7) d t]$

Step 1.1.3

Factor $t$ out of $t \cdot t^{3} + t \cdot 7$ .

$\frac{d}{d x} [\int \sqrt{t (t^{3} + 7)} (4 t^{3} + 7) d t]$

Step 1.2

Apply the distributive property.

$\frac{d}{d x} [\int \sqrt{t (t^{3} + 7)} (4 t^{3}) + \sqrt{t (t^{3} + 7)} \cdot 7 d t]$

Step 1.3

Reorder.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [\int 4 \sqrt{t (t^{3} + 7)} t^{3} + \sqrt{t (t^{3} + 7)} \cdot 7 d t]$

Step 1.3.2

Move $7$ to the left of $\sqrt{t (t^{3} + 7)}$ .

$\frac{d}{d x} [\int 4 \sqrt{t (t^{3} + 7)} t^{3} + 7 \sqrt{t (t^{3} + 7)} d t]$

Step 2

Since $\int 4 \sqrt{t (t^{3} + 7)} t^{3} + 7 \sqrt{t (t^{3} + 7)} d t$ is constant with respect to $x$ , the derivative of $\int 4 \sqrt{t (t^{3} + 7)} t^{3} + 7 \sqrt{t (t^{3} + 7)} d t$ with respect to $x$ is $0$ .

$0$