Techniques of Integration

The Techniques of Integration represent a series of strategies employed to calculate complex integrals within the realm of calculus. These strategies encompass substitution, integration by parts, trigonometric integrals, trigonometric substitution, partial fractions, and improper integrals. Proficiency in these methods is crucial for tackling and solving sophisticated mathematical challenges.

Integration by Parts

Evaluate. (Be sure to check by differentiating!)

\int (4 t^{3} - 7) t^{2} d t

Determine a change of variables from

t

u

. Choose the correct answer below. A.

u = 4 t - 7

u = t^{2} - 7

u = 4 t^{3} - 7

u = t^{2}

Write the integral in terms of

u

\int (4 t^{3} - 7) t^{2} d t = \int (◻) d u

(Type an exact answer. Use parentheses to clearly denote the argument of each function.) Evaluate the integral.

\int (4 t^{3} - 7) t^{2} d t =

(Type an exact answer. Use parentheses to clearly denote the argument of each function.)

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Trigonometric Integrals

Evaluate the indefinite integral

\int 8 \sin^{6} x \cos x d x =

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Trigonometric Substitution

Find

\frac{d y}{d r}

for

y = \int_{0}^{r} \sqrt{11 + 12 t^{2}} d t

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Integration by Partial Fractions

Evaluate the integral

\int_{1}^{\sqrt{2}} \frac{s^{2} + \sqrt{s}}{s^{2}} d s

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Techniques of Integration

Integration by Parts

Trigonometric Integrals

Trigonometric Substitution

Integration by Partial Fractions

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