Integration by substitution Quiz

1. Find $\int2xe^{x^2}\,\mathrm{d}x$.

• A. $2e^{x^2}+c$
• B. $x^2e^{x^2}+c$
• C. $e^{x^2}+c$
• D. $\frac{e^{x^2}}{2x}+c$

2. Find $\int\cos x\sin^4x\,\mathrm{d}x$.

• A. $\frac{\sin^5x}{5}+c$
• B. $\frac{\cos^5x}{5}+c$
• C. $-\frac{\sin^5x}{5}+c$
• D. $4\sin^3x\cos x+c$

3. With $u=3x+2$, what is $\mathrm{d}x$?

• A. $3\,\mathrm{d}u$
• B. $\mathrm{d}u$
• C. $\frac{\mathrm{d}u}{x}$
• D. $\frac13\mathrm{d}u$

4. For $\int_0^1x\sqrt{1+x^2}\,\mathrm{d}x$ with $u=1+x^2$, what are the limits for $u$?

• A. $0$ and $1$
• B. $1$ and $2$
• C. $1$ and $5$
• D. $0$ and $2$

5. Evaluate $\int_0^1\frac{2x}{x^2+1}\,\mathrm{d}x$.

• A. $1$
• B. $\ln1$
• C. $\ln2$
• D. $\frac12\ln2$