Integration by substitution

Evaluate each of the following indefinite integrals. Check each antiderivative that you find by differentiating.

1. $\displaystyle \int \sin(8-3x) \, dx$

2. $\displaystyle \int \sec^2 (4x) \, dx$

3. $\displaystyle \int \frac{1}{11x - 9} \, dx$

4. $\displaystyle \int \csc(2x+1) \cot(2x+1) \, dx$

5. $\displaystyle \int \frac{1}{\sqrt{1-16x^2}}\, dx$

6. $\displaystyle \int 5^{-x}\, dx$

Evaluate each of the following indefinite integrals by using these steps:

• Find two functions within the integrand that form (up to a possible missing constant) a function-derivative pair;

• Make a substitution and convert the integral to one involving $u$ and $du$;

• Evaluate the new integral in $u$;

• Convert the resulting function of $u$ back to a function of $x$ by using your earlier substitution;

• Check your work by…

Evaluate each of the following definite integrals exactly through an appropriate $u$-substitution.

1. $\displaystyle \int_1^2 \frac{x}{1 + 4x^2} \, dx$

2. $\displaystyle \int_0^1 e^{-x} (2e^{-x}+3)^{9} \, dx$

3. $\displaystyle \int_{2/\pi}^{4/\pi} \frac{\cos\left(\frac{1}{x}\right)}{x^{2}} \,dx$

This problem centers on finding antiderivatives for the basic trigonometric functions other than $\sin(x)$ and $\cos(x)$.

1. Consider the indefinite integral $\int \tan(x) \, dx$. By rewriting the integrand as $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and identifying an appropriate function-derivative pair, make a $u$-substitution and hence evaluate $\int \tan(x) \, dx$.

2. In a similar way, evaluate $\int \cot(x) \, dx$.

3. Consider the indefinite integral . Evaluate this integral using the substitution $u = \sec(x) + \tan(x)$.

4. Simplify the in…

Consider the indefinite integral $\int x \sqrt{x-1} \, dx$.

1. At first glance, this integrand may not seem suited to substitution due to the presence of $x$ in separate locations in the integrand. Nonetheless, using the composite function $\sqrt{x-1}$ as a guide, let $u = x-1$. Determine expressions for both $x$ and $dx$ in terms of $u$.

2. Convert the given integral in $x$ to a new integral in $u$.

3. Evaluate the integral in (b) by noting that $\sqrt{u} = u^{1/2}$ and observing t…

Consider the indefinite integral $\int \sin^3(x) \, dx$.

1. Explain why the substitution $u = \sin(x)$ will not work to help evaluate the given integral.

2. Recall the Fundamental Trigonometric Identity, which states that $\sin^2(x) + \cos^2(x) = 1$. By observing that $\sin^3(x) = \sin(x) \cdot \sin^2(x)$, use the Fundamental Trigonometric Identity to rewrite the integrand as the product of $\sin(x)$ with another function.

3. Explain why the substitution $u = \cos(x)$ now provides a possible way to evaluate the integral i…