5.5 The Method of Substitution - Introduction to Calculus

The Method of Substitution

The idea behind the method of substitution is to change a difficult integral in terms of one variable into an easier integral in terms of some other variable using a substitution.

THEOREM

The Method of Substitution

\displaystyle \begin{aligned}\int f'(u(x)) \ {du \over dx} \, dx = \int f'(u) \ du\end{aligned}

The method of substitution.

\displaystyle \int (6x + 4)(3x^2 + 4x)^5 \ dx

Calculate using the method of substitution.

(a)

\displaystyle \int \sin(x)e^{5\cos(x)} \ dx

(b)

\displaystyle \int {x \over (5x + 7)^3} \ dx

(c)

\displaystyle \int x\sec(3x^2)\tan(3x^2) \ dx

(d)

\displaystyle \int {-{1 \over t^2} + 1 \over \sqrt{{1 \over t} + t}} \ dt

(e)

\displaystyle \int (1 + 900x)^{1/15000} \ dx

(f)

\displaystyle \int \sin(\theta)(\cos^3\theta - \cos^5\theta) \ d\theta

(g)

\displaystyle \int {7x \over 4x^2 + 9} \ dx

(h)

\displaystyle \int (2x + 5) \cdot \sqrt[3]{3x + 1} \ dx

QUESTION

\displaystyle u=5x+7

, which substitution is correct for

\displaystyle x\,dx

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The Method of Substitution

The idea behind the method of substitution is to change a difficult integral in terms of one variable into an easier integral in terms of some other variable using a substitution.

THEOREM

The Method of Substitution

\displaystyle \begin{aligned}\int f'(u(x)) \ {du \over dx} \, dx = \int f'(u) \ du\end{aligned}

The method of substitution.

\displaystyle \int (6x + 4)(3x^2 + 4x)^5 \ dx

Calculate using the method of substitution.

(a)

\displaystyle \int \sin(x)e^{5\cos(x)} \ dx

(b)

\displaystyle \int {x \over (5x + 7)^3} \ dx

(c)

\displaystyle \int x\sec(3x^2)\tan(3x^2) \ dx

(d)

\displaystyle \int {-{1 \over t^2} + 1 \over \sqrt{{1 \over t} + t}} \ dt

(e)

\displaystyle \int (1 + 900x)^{1/15000} \ dx

(f)

\displaystyle \int \sin(\theta)(\cos^3\theta - \cos^5\theta) \ d\theta

(g)

\displaystyle \int {7x \over 4x^2 + 9} \ dx

(h)

\displaystyle \int (2x + 5) \cdot \sqrt[3]{3x + 1} \ dx

QUESTION

\displaystyle u=5x+7

, which substitution is correct for

\displaystyle x\,dx

The Method of Substitution